UNIVERSIDAD COMPLUTENSE DE MADRID

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1 UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS DEPARTAMENTO DE MATEMÁTICA APLICADA TESIS DOCTORAL Nonlocal diffusion problems Problemas de difusión no local MEMORIA PARA OPTAR AL GRADO DE DOCTORA PRESENTADA POR Silvia Sastre Gómez Director Aníbal Rodríguez Bernal Madrid, 214 Silvia Sastre Gómez, 214

2 Universidad Complutense de Madrid Facultad de Ciencias Matemáticas Departamento de Matemática Aplicada Nonlocal diffusion problems Problemas de difusión no local Memoria para optar al grado de Doctor en Matemáticas presentada por: Silvia Sastre Gómez Bajo la dirección de Aníbal Rodríguez Bernal Madrid 214

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4 A mis padres.

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6 Agradecimientos En primer lugar me gustaría comenzar dando gracias a mi director de tesis Aníbal Rodríguez Bernal, por toda la ayuda recibida durante estos años, matemática y no tan matemática. Sobre todo por su infinita paciencia y por su entusiasmo, que muchas veces era un gran aliento para poder continuar adelante. Todas las discusiones durante estos años me han hecho crecer un poco más como matemática y como persona. También querría agradecer al Departamento de Matemática Aplicada por ofrecerme las comodidades para poder estudiar cada día en la facultad. Por otro lado, agradezco a la Universidad Complutense de Madrid y al Ministerio de Eduación por concederme la beca que ha hecho posible que pudiese desarrollar esta tesis. Muchas gracias al profesor Emmanuel Chasseigne, por toda la ayuda prestada durante mi estancia en Tours, tanto matemática como personal. Y por todas las veces que durante estos años me ha ayudado. También agradecer al profesor James Robinson su muy agradable acogida durante mi estancia en Warwick. Muchas gracias por las conversaciones matemáticas, por las comidas y por su hospitalidad. Me gustaría hacer un hueco en estos párrafos a Julia y Ale que hicieron de mi estancia en Warwick, una época que recordaré con mucho cariño. Agradecer a mi familia y a mis amigos el apoyo recibido durante estos años. Y sobretodo agradecer a mis compañeros, sin los cuáles, todo habría sido mucho menos llevadero. Muchas gracias por todos los cafés por las mañanas, discusiones matemáticas, las comidas, meriendas, poleos, chistes buenos (y no tan buenos), que han hecho que durante este periodo todos los días hayan merecido una sonrisa. Gracias a Alba, Simone, Carlos P., Alfonso, Carlos Q., Manuel, María, Luis F., Marcos, Edwin, Luis H., Andrea, Espe, Nadia, Giovanni, Nacho, Javi, Alicia, Álvaro, Diego, y muchos otros. Por último quiero dejar estas últimas lineas para agradecer a Diego, que me ha ayudado muchísimo tanto matemáticamente como personalmente. Gracias por tu apoyo, confianza, paciencia y ánimo.

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8 Contents Resumen Introduction ix xxi 1 Nonlocal diffusion on metric measure spaces Metric measure spaces Function spaces in a metric measure space Some examples of metric measure spaces Manifolds, Multi-structures and other metric measure spaces Nonlocal diffusion problems The linear nonlocal diffusion operator Properties of the operator K Regularity of K J Regularity of convolution operators Compactness of K J Positiveness of the operator K J The adjoint operator of K J Spectrum of K J The multiplication operator hi Green s formulas for K J h I Spectrum of the operator K hi The linear evolution equation Existence and uniqueness of solution of (3.1) The solution u of (3.1) is positive if the initial data u is positive Asymptotic regularizing effects The Riesz projection and asymptotic behavior Asymptotic behaviour of the solution of the nonlocal diffusion problem Nonlinear problem with local reaction Existence, uniqueness, positiveness and comparison of solutions with a globally Lipschitz reaction term Existence and uniqueness of solutions, with locally Lipchitz f vii

9 4.3 Asymptotic estimates Extremal equilibria Instability results for nonlocal reaction diffusion problem Nonlocal reaction-diffusion equation The nonlocal reaction term Existence, uniqueness, positiveness and comparison of solutions with a nonlinear globally Lipchitz term Existence and uniqueness of solutions, with a nonlinear locally Lipschitz term Asymptotic estimates and extremal equilibria Attractor A nonlocal two phase Stefan problem Basic theory of the model Existence, positiveness and comparison of solutions Free boundaries First results concerning the asymptotic behavior Asymptotic behavior when the positive and the negative part of the temperature do not interact Formulation in terms of the Baiocchi variable A nonlocal elliptic biobstacle problem Asymptotic limit for general data Solutions losing one phase in finite time A theoretical result Sufficient conditions to lie above level 1 in finite time Appendix A: L p spaces 163 Appendix B: Nemitcky operators 165 Bibliography 169 viii

10 Resumen Introducción La difusión es un proceso natural por el que, por ejemplo, la materia es transportada de un lugar a otro como resultado del movimiento molecular aleatorio. El experimento clásico que ilustra este proceso es aquel en el que se coloca una gota de tinta en un recipiente lleno de agua, y la tinta tiende a extenderse por todo el recipiente y la solución aparece coloreada de manera uniforme. (Existen experimentos más refinados para asegurar que no haya convección). Los modelos de difusión aparece en diferentes áreas como biología, termodinámica, medicina, e incluso economía. En biología, existen modelos que estudian la dinámica poblacional, i.e., cambios a corto y largo plazo, en el tamaño y edad de la población, y procesos medioambientales y biológicos que influyen en esos cambios. La dinámica poblacional se enfrenta con la forma en la que la población se ve afectada por la tasas de natalidad y mortalidad, y por la inmigración y la emigración. En medicina, los modelos de difusión se usan, por ejemplo, para describir crecimientos tumorales. En termodinámica, la ecuación del calor modela la conducción del calor, esto es cuando un objeto está a diferente temperatura que otro cuerpo, o que a su alrededor, el calor fluye de manera que el cuerpo y sus alrededores alcanzan la misma temperatura. En economía, la difusión modela las fluctuaciones del mercado de valores, usando movimientos brownianos. Existen dos manera de introducir la noción de difusión: con la aproximación fenomenológica, comenzando con las leyes de difusión de Fick, o con la aproximación física o atómica, considerando movimientos aleatorios de la difusión de partículas. Primero, introducimos las leyes que rigen los procesos de difusión: Las leyes de Fick. Esta leyes relacionan el flujo difusivo con la concentración bajo la hipótesis de estado estacionario. Ésta postula que el flujo se mueve de regiones con alta concentración hacia regiones con baja concentración, con una magnitud que es proporcional al gradiente de concentración. Entonces en dimensión 1, tenemos F = D u x, (1) donde F es el flujo de difusión, u es la concentración de la substancia que se difunde, y D es el coeficiente de difusión. Por otro lado, por la ley de Fick y la conservación de la masa en ausencia de reacciones químicas: u t + F =. (2) x ix

11 Entonces, por (1) y (2), obtenemos la segunda ley de Fick que predice cómo la difusión provoca cambios en la concentración con el tiempo: u t = D 2 u x 2. (3) Para el caso de la difusión en dos o más dimensiones, la segunda ley de Fick viene dada por u t = D u. (4) Los modelos de difusión local vienen dados por (4) más condiciones iniciales y de frontera, necesarias para completar el modelo. Desde el punto de vista atómico, la difusión es considerada el resultado del movimento aleatorio, (random walk) de partículas difusivas. En difusión molecular, las moléculas que se mueven, son propulsadas por energía térmica. El movimiento aleatorio de pequeñas partículas en suspensión en un fluido fue descubierto en 1827 por Robert Brown, y la teoría de movimiento Browniano y el punto de vista atómico de la difusión, fue desarrollado por Albert Einstein en 195. Otro tipo de modelos de difusión son los modelos de difusión no local. Estos modelos pueden derivarse de variaciones de procesos de salto (ver por ejemplo [35]). Consideremos una única especie en un hábitat N-dimensional donde se asume que la población se puede modelar por una función u(x, t), que es la densidad en x en tiempo t. Un modelo continuo para la dinámica poblacional de especies se puede derivar considerando con detalle una discretización en espacio y tiempo, y después haciendo tender los intervalos de espacio y tiempo a cero. En particular, la derivación clásica del laplaciano,, (4) por movimientos aleatorios, se tiene asumiendo una distribución binomial. Sin embargo, en el caso de la difusión no local, consideramos cualquier tipo de distribución. A continuación reproducimos la derivación del modelo no local para el caso N = 1. Consideramos que el habitat es R. Primero, dividimos en intervalos contiguos, cada uno de longitud x, y discretizamos el tiempo en pasos de tamaño t. Sea u(i, t) la densidad de individuos en la posición i en tiempo t. Queremos derivar el cambio en el número de individuos en esta posición durante el siguiente intervalo de tiempo. La primera hipótesis es que la tasa a la que los individuos salen de i para llegar a j es constante. Por tanto, el número total de individuos saliendo de i a j debería ser proporcional a: la población en el intervalo i, que es u(i, t) x; el tamaño del lugar al que llegan, que es x; y la cantidad de tiempo durante el cuál el tránsito se está midiendo, t. Sea J(j, i) la constante proporcional, entonces, el número de individuos saliendo de i durante el intervalo de tiempo [t, t + t] es M j= M j i J(j, i)u(i, t)( x) 2 t. (5) Durante este mismo intervalo de tiempo, el número de llegadas a i desde otros lugares es M j= M j i J(i, j)u(j, t)( x) 2 t. (6) x

12 Combinando (5) y (6), deducimos que la densidad de población en i en tiempo t + t viene dado por u(i, t + t) x = u(i, t) x + M j= M j i entonces, dividiendo (7) entre x, obtenemos u(i, t + t) = u(i, t) + M j= M j i J(i, j)u(j, t)( x) 2 t J(i, j)u(j, t) x t M j= M j i M j= M j i J(j, i)u(i, t)( x) 2 t, (7) J(j, i)u(i, t) x t. (8) Entonces, tendiendo t y x en (8), tenemos ( ) u t (x, t) = J(x, y)u(y, t) J(y, x)u(x, t) dy. (9) Ahora, reinterpretamos (9), con R. Asumimos que J(x, y) es una función positiva definida en que representa la densidad de probabilidad de saltar de y a x, y u(x, t) es la densidad de población en el punto x en tiempo t, entonces J(x, y)u(y, t)dy es la tasa a la que los individuos llegan a x desde otros lugares y. Como hemos asumido que J es la densidad de probabilidad, y J está definida en, entonces J(x, y)dy = 1. En particular, u(x, t) = J(x, y)dy u(x, t) es la tasa a la que los individuos salen de x a otras posiciones y. Entonces, podemos escribir la ecuación (9) con condición inicial u, como u t (x, t) = J(x, y)u(y, t)dy u(x, t), x, (1) u(x, ) = u (x), x. Este problema y variantes de él, ha sido previamente usado para modelar procesos de difusión, por ejemplo en [2], [18], [27], y [35]. Este modelo permite tener en cuenta interacciones a corta (short-range) y larga (long-range) distancia, y es posible generalizar el problema (1), para R N, o incluso espacios medibles más generales, (ver Capítulo 1). El modelo (1) se llama modelo de difusión no local, pues la difusión de la densidad u en x en tiempo t no depende únicamente de u(x, t), sino que depende de todos los valores de u en un entorno de x, a través del término de convolución J(x, y)u(y, t)dy. Objetivos Ahora, fijemos un conjunto abierto R N. Para problemas locales como (4), las dos condiciones de contorno más habituales son la de Neumann y la de Dirichlet. La ecuación del calor local con condición frontera Neumann, viene dada por u t (x, t) = u(x, t), x, t >, u (x, t) =, x, t >, (11) ν u(x, ) = u (x), x, xi

13 donde ν denota la normal exterior a la frontera, y u ν = modela que los individuos no entren ni salgan de. Un problema no local análogo definido en el abierto R N, propuesto en [18], viene dado por u t (x, t) = u(x, ) = u (x), J(x, y) (u(y, t) u(x, t)) dy = donde J : R N R N R, con R N J(x, y)dy = 1, y denotamos h (x) = J(x, y)dy, x. J(x, y)u(y, t)dy h (x)u(x, t) En (12), la integral está definida sobre, entonces, este modelo asume que los individuos no entran ni salen de, y la difusión tiene lugar sólo dentro de. Además, (12) comparte con el problema local (11), que las constantes son equilibrios. (12) es Pro otro lado, la ecuación del calor local con condición de frontera Dirichlet homogénea u t (x, t) = u(x, t), x, t >, u(x, t) =, x, t >, u(x, ) = u (x), x. En este caso, u es cero en la frontera del hábitat. Un problema no local análogo propuesto en [18] con R N abierto, viene dado por u t (x, t) = J(x, y)u(y, t)dy u(x, t), x, t >, R N u(x, t) =, x /, t >, u(x, ) = u (x), x. En este modelo la difusión tiene lugar en todo R N, y u = fuera de. Entonces, este problema modela el caso en que los individuos mueren cuando salen del hábitat, y R J(x, y)u(y, t)dy = N J(x, y)u(y, t)dy. Entonces, la ecuación (13), es u t (x, t) = J(x, y)u(y, t)dy u(x, t). Los problemas (9), (1), (12) y (13) se pueden unificar considerando el problema no local u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t), x, t >, u(x, ) = u (x), x, con h definida en. Éste es el tipo de problemas lineales no locales que vamos a estudiar. (13) (14) xii

14 Sobre problemas no lineales, introducimos los modelos no locales de reacción-difusión, añadiendo un término de reacción local f(x, u(x, t)) al modelo de difusión (14), u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t) + f(x, u(x, t)), x, t >, (15) u(x, ) = u (x), x, donde f : R R. Éste modelo fue considerado en [35], donde (15) modela la dinámica poblacional de las especies, y f denota la tasa de reproducción en x de una densidad de población u(x, t), que tiene en cuenta el número de individuos nuevos en x en tiempo t. También consideramos modelos de reacción-difusión, con difusión no local y reacción no local. El problema viene dado por u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t) + f(x, u)(, t), x, t >, (16) u(x, ) = u (x), x, pero ahora f : L 1 () R es un término no local. Éste modelo ha sido previamente considerado en [3]. Los problemas con difusión local y reacción no local han sido considerados en [11], donde el término de reacción no local tiene en cuenta la saturación no local o los efectos de competición no local. Otro tipo de modelos de difusión no local, es el que aparece en [12, 19], dado por u t (x, t) = J(x, y)(γ(u(y, t)) Γ(u(x, t)))dy, x R N, t >, (17) R N donde Γ(u) = sign(u) ( u 1 ). Este problema se llama problema de Stefan no local. Modela la distribución de la temperatura y la entalpía en una fase de transición entre diferentes + estados, por ejemplo, el cambio de fases entre hielo y agua. Actualmente, existe un gran interés en el estudio de la difusión en dominios no regulares. Existen varios intentos de generalizar el operador laplaciano a espacios no regulares: las formas de Dirichlet (Dirichlet forms), ayudan a describir procesos de salto que se pueden definir en espacios no regulares. Pro tanto, es posible definir ecuaciones diferenciales en espacios no regulares, como pueden ser los fractales. Con esta teoría, llamada Análisis en fractales, se extienden conceptos como el laplaciano, las funciones de Green, núcleos de calor, (ver [9, 37, 5]). Por otro lado, los modelos de difusión no local, como (14), (15), (16) se pueden definir en espacios métricos de medida (ver Capítulo 1), pues simplemente necesitamos considerar la densidad de probabilidad de saltar de un punto a otro de, que viene dada por J(x, y). Y este tipo de densidad se puede definir en un espacio métrico de medida general. Lo que nos permite estudiar la difusión en espacios muy diferentes como: grafos, (usados para modelar estructuras complicadas en química, biología molecular o electrónica, incluso pueden representar circuitos eléctricos en computadoras digitales); variedades compactas; multiestructuras xiii

15 compuestas por conjuntos compactos de diferentes dimensiones, (por ejemplo un conjunto de Dumbbell, donde es necesario considerar una perturbación del dominio para estudiar problemas de difusión local, como se puede ver en [3], mientras que en los problemas de difusión no local podremos estudiar el problema directamente en el dominio); o incluso conjuntos fractales como el triángulo de Sierpinski. Resultados Centrémonos en lo que será hecho a lo largo de este trabajo. Como mencionamos arriba, en esta tesis estudiamos problemas de difusión no locales generales. Sea µ una medida, y d una métrica definida en, consideramos un espacio métrico de medida (, µ, d), que se introduce en el Capítulo 1. Primero, consideramos el problema de difusión lineal no local dado por { u t (x, t) = (K hi)(u)(x, t), x, t > (18) u(x, ) = u (x), x, donde es el operador integral, y K(u)(x, t) = J(x, y)u(y, t)dy hi(u)(x, t) = h(x)u(x, t) es el operador multiplicación con h L () o en C b (), donde C b () son las funciones continuas y acotadas definidas en. No asumiremos, a no ser que se diga explícitamente, que J(x, y)dy = 1. Una función que será importante a lo largo de este trabajo es h (x) = J(x, y)dy, que no es necesariamente igual a la identidad. Para estudiar el problema lineal (18), en el Capítulo 2, primeramente realizaremos un estudio completo del operador lineal K hi, estudiando los espacios donde el operador está definido, la compacidad y el espectro de K y hi de manera separada. Después en el Capítulo 3, nos concentramos en la existencia y unicidad de soluciones de (18); en las propiedades de monotonía de las soluciones en X = L p (), con 1 p o X = C b (). Recuperamos y generalizamos el estudio de existencia y unicidad de soluciones de (18), con h = h o h = Id. Lo cuál ha sido hecho en L 1 () en [2, 18], considerando un dominio R N abierto. A continuación, estudiamos el comportamiento asintótico de las soluciones cuando el tiempo se va a infinito. Probamos que si σ X (K hi) es la unión de dos conjuntos cerrados disjuntos σ 1 y σ 2 con Re(σ 1 ) δ 1, Re(σ 2 ) δ 2, con δ 2 < δ 1, entonces el comportamiento asintótico de la solución de (18) en X está descrito por la proyección de Riesz de K hi correspondiente a σ 1. Probamos también que la proyección de Riesz y la proyección de Hilbert son iguales. Además, aplicamos este resultado a los casos particulares del problema de difusión no local (18) con h constante y h = h. En particular, recuperamos y generalizamos el resultado xiv

16 en [18], para X = L p (), con 1 p o X = C b (), mientras que en [18], los autores obtienen el resultado en L 2 () si el dato inicial está en L 2 (), y en L () si el dato inicial está en C(), considerando R N un conjunto abierto. El estudio del problema (18) nos lleva a la conclusión de que la ecuación (18) comparte algunas propiedades con la ecuación clásica del calor. En particular, ambas tienes Principio Débil y Fuerte del Máximo, cuando J satisface hipótesis de positividad, pero no comparten el efecto regularizante, como se indica en [27], para el caso = R N. Esto ocurre porque la solución de (18) conserva las singularidades de los datos iniciales. Sin embargo, hemos podido probar que el semigrupo S(t) de (18) satisface que S(t) = S 1 (t) + S 2 (t), con S 1 (t) que converge a mientras t va a infinito en X, y S 2 (t) es compacta, entonces S(t) es asintóticamente compacta, (asymptotically smooth), de acuerdo con la definición en [32, p. 4]. En el Capítulo 4, consideramos una ecuación de reacción-difusión no local, con término de reacción no lineal, y trabajamos con el problema { u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)), x, t > (19) u(x, t ) = u (x), x, con f : R R, y dato inicial u L p (). La función f(x, s) se asumirá que es localmente Lipschitz en la variable s R, uniformemente con respecto a x. Existe una amplia literatura sobre el estudio de problemas de reacción-difusión locales { u t (x, t) = u(x, t) + f(x, u(x, t)), x, t >, (2) u(x, t ) = u (x), x. Existencia, unicidad y resultados de comparación de la soluciones de (43) con término de reacción no lineal localmente Lipschitz, f, como en (19) satisfaciendo condiciones de signo son conocidas, ver por ejemplo [47, 4]. Los argumentos usados para el problema (2) son esencialmente argumentos de punto fijo, pero no podemos usar estos argumentos para el problema no local (19), porque el semigrupo lineal S(t) asociado a (18) no regulariza. Probamos primero la existencia para la ecuación (19) con f globalmente Lipschitz, y después probamos la existencia con f localmente Lipschitz satisfaciendo condiciones de signo con argumentos de sub-supersolución, en X = L p (), con 1 p o X = C b (). Por tanto, recuperamos y generalizamos los resultados de existencia y unicidad de las soluciones de (19) en [8], donde R N y el dato inicial está en C(). Observamos que en [3], los autores estudian los exponentes de Fujita para (19), que coinciden con los clásicos de (2). También estudiamos el comportamiento asintótico de las soluciones de (19). En [44], bajo condiciones de signo en el término no lineal, los autores prueban la existencia de dos equilibrios maximales de (2), con R N un dominio acotado y diferentes tipos de condiciones de contorno. También prueban que la dinámica asintótica de las soluciones entra entre estos equilibrios maximales, uniformemente en espacio, para conjuntos acotados de datos iniciales. Como consecuencia, obtienen una cota del atractor global para las ecuaciones de reaccióndifusión locales, (2). Por otro lado, nosotros probamos la existencia de dos equilibrios maximales ordenados ϕ m y ϕ M (uno minimal y otro maximal), para el problema (19), y toda la dinámica asintótica xv

17 de las soluciones de (19) con datos iniciales acotados, entra entre los dos equilibrios maximales ϕ m y ϕ M, cuando el tiempo va a infinito en L p (), para todo 1 p <. Además éstos mismos equilibrios extremales, ϕ m y ϕ M, son cotas de cualquier límite débil en L p (), con 1 p <, de las soluciones de (19) con datos iniciales u en L p (). Observamos que para el problema no local (19), obtenemos resultados más débiles que para el problema local (2) de nuevo, por la falta de regularización del semigrupo asociado al problema lineal no local. Después de estudiar el comportamiento asintótico, discutimos la existencia y estabilidad de equilibrios del problema u t (x, t) = J(x, y)u(y, t)dy h (x) u(x, t) + f(u(x, t)), x, t > (21) u(x, ) = u (x), x, Sea F el operador de Nemitcky asociado a la función f, tal que F (u)(x, t) = f(u(x, t)). Como F : L p () L p () no es diferenciable (ver Apéndice B), y el semigrupo asociado al problema lineal no local (18) no regulariza, entonces el Principio de Estabilidad linealizada falla. Sin embargo, bajo hipótesis en la convexidad de la función f, probamos que la estabilidad/inestabilidad respecto a la linealización, implica la estabilidad/inestabilidad de los equilibrios del problema no lineal (21). También probamos que cualquier equilibrio no constante de (21) es, si existe, inestable cuando f es convexa. En [16], [14] y [4], los autores prueban resultados similares para el problema de reacción-difusión local (2) con condición de frontera Neumann. En [14] y [4], los autores también prueban que si es un dominio convexo, entonces cualquier equilibrio no constante, es inestable, es decir, no existen patrones (patterns). Hasta donde nosotros sabemos, este resultado no ha sido probado para el problema no local (21), y las técnicas que se usan para el problema local, no parecen ser útiles para probar la no existencia de patrones si el dominio es convexo. Existe un gran interés en el estudio de existencia y estabilidad de equilibrios del problema (19). En [8], los autores estudian la estabilidad de equilibrios positivos con dato inicial en C(). En particular, prueban que bajo hipótesis en el espectro del operador lineal K J, existe un único equilibrio no negativo asintóticamente estable en C() +. En el Capítulo 5, estudiamos problemas de reacción-difusión con ambos términos no locales, i.e., consideramos el problema { u t (x, t) = (K hi)(u)(x, t) + f(x, u)(, t), x, t > (22) u(x, ) = u (x), x, donde (K hi)(u) es el término de difusión no local, y f : L 1 () R es el término de reacción no local, y está definido como sigue f = g m, donde g : R R es una función no lineal, y m : L 1 () R es la media de u en la bola xvi

18 de radio δ > y centro x, definido como m(x, u(, t)) = 1 u(y, t)dy. µ(b δ (x)) B δ (x) Primero derivamos una teoría completa de existencia y unicidad para el problema (22), en X = L p (), con 1 p o X = C b (), con g globalmente Lipschitz. El problema (22) no tiene propiedades de comparación en general. Por tanto, damos resultados de comparación para el problema (22), con g globally Lipschitz, con constante Lipschitz suficientemente pequeña en comparación con J, usando argumentos de punto fijo. Si g es localmente Lipschitz, y satisface condiciones de signo, entonces probamos existencia y unicidad de solución para el problema (22), con término no lineal g, tal que la constante de Lipschitz de g k es suficientemente pequeña en comparación con J, donde g k es la función truncada en k asociada a g. De hecho, la existencia y unicidad será probada para datos iniciales en L (), tales que u L () k. Además, probaremos propiedades de comparación para las soluciones de (22) con g y u satisfaciendo las condiciones de arriba. Probamos también que la dinámica asintótica de las soluciones de (22) con g globalmente Lipschitz, entra entre los dos equilibrios extremales ϕ m y ϕ M, como hacemos para los problemas de reacción-difusión no local (19). Además, si suponemos que el promedio en la bola de radio delta es continuo, entonces probamos que los equilibrios ϕ m, ϕ M C b () y la dinámica asintótica de las soluciones de (22) entra entre ϕ m y ϕ M uniformemente en conjuntos compactos de. Otra ventaja de este modelo (22), con reacción no local respecto al problema de reacción difusión no local (19), es que el término de reacción no local F : L p () L p () es compacto, y probamos que el semigrupo asociado a (22) es asintóticamente compacto, y entonces usamos [32, Theorem ], para probar la existencia de un atractor global para el semigrupo de (22). En el Capítulo 6, estudiamos el problema de Stefan de dos-fases no local en R N u t = J(x y)v(y)dy v, where v = Γ(u), R N u(, ) = f, (23) donde J es un núcleo de convolución no negativo suave, u es la entalpía y Γ(u) = sign(u) ( u 1 ) +. El problema de Stefan es un problema no lineal de frontera móvil cuyo objetivo es describir la distribución de la temperatura y la entalpía en una fase de transición entre diferentes estados. La historia del problema comenzó con Lamé y Clapeyron [39], y después con Stefan, en [49]. Para el modelo local se puede ver por ejemplo las monografías [17] y [54] para las fenomenológicas y modelización, [23], [41], [45] y para los aspectos matemáticos del modelo [53]. El modelo principal usa la ecuación local u t = v, v = Γ(u), pero recientemente, una versión no local del problema de Stefan de una-fase fue introducido en [12], que es equivalente a (23) en el caso de soluciones no negativas, y Γ(u) viene dada por Γ(u) = ( u 1 ) +. xvii

19 Este nuevo modelo matemático es interesante desde el punto de vista de la física, pues a escala intermedia (mesoscópica), explica por ejemplo la evolución de mushy regions (regiones que son un estado intermedio entre hielo y agua). Nosotros estudiamos la existencia, unicidad y comparación en la linea de los capítulos anteriores, y estudiamos el comportamiento asintótico en el espíritu de [12], pero para soluciones que cambian de signo, lo que presenta retos muy difíciles sobre el comportamiento asintótico. Aunque no damos un estudio completo del comportamiento asintótico, que parece ser bastante difícil, damos condiciones suficientes que garanticen la identificación del límite cuando el tiempo tiende a infinito. Conclusiones Los modelos de difusión no local, se pueden platear en espacios métricos de medida (ver Capítulo 1). Lo que nos permite estudiar procesos de difusión en espacios muy diferentes como: grafos, multiestructuras compuestas por conjuntos compactos de diferentes dimensiones, o incluso conjuntos fractales como el triángulo de Sierpinski. El estudio del problema lineal (18) nos lleva a la conclusión de que la ecuación (18) comparte algunas propiedades con la ecuación clásica del calor. En particular, ambas tienen Principio Débil y Fuerte del Máximo, cuando J satisface hipótesis de positividad, pero no comparten el efecto regularizante. Esto ocurre porque la solución de (18) carga con las singularidades de los datos iniciales. Sin embargo, hemos podido probar que el semigrupo S(t) de (18) satisface que S(t) = S 1 (t) + S 2 (t), con S 1 (t) que converge a mientras t va a infinito en X, y S 2 (t) es compacta, entonces S(t) es asintóticamente compacta, (asymptotically smooth), de acuerdo con la definición en [32, p. 4]. Para el problema no local (19), probamos la existencia de dos equilibrios maximales ordenados ϕ m y ϕ M, y toda la dinámica asintótica de las soluciones de (19) con datos iniciales acotados, entra entre los dos equilibrios maximales ϕ m y ϕ M, cuando el tiempo va a infinito en L p (), para todo 1 p <. Además éstos mismos equilibrios extremales, ϕ m y ϕ M, son cotas de cualquier límite débil en L p (), con 1 p <, de las soluciones de (19) con datos iniciales u en L p (). Estos resultados son más débiles que para el problema local (2), debido a la falta de regularización del semigrupo asociado al problema lineal no local. Como F : L p () L p () no es diferenciable (ver Apéndice B), y el semigrupo asociado al problema lineal no local (18) no regulariza, entonces el Principio de Estabilidad linealizada falla. Sin embargo, bajo hipótesis en la convexidad de la función f, probamos que la estabilidad/inestabilidad respecto a la linealización, implica la estabilidad/inestabilidad de los equilibrios del problema no lineal (21). Además si f es cóncava, probamos que no existen patrones para el problema (21). El problema (22) puede no cumplir las propiedades de comparación. Por tanto, damos resultados de comparación para el problema (22), donde g tiene una constante Lipschitz suficientemente pequeña en comparación con J. xviii

20 Probamos la existencia de equilibrios maximales ϕ m y ϕ M para el problema (22). Y como el término de reacción no local regulariza, probamos que la dinámica asintótica de las soluciones de (22) entra entre ϕ m y ϕ M uniformemente en conjuntos compactos de. Además podemos probar que el semigrupo asociado al problema (22) es asintóticamente regular, y por tanto, probamos la existencia de un atractor global para el semigrupo del problema. Para el problema no local de Stefan de dos fases (23) estudiamos el comportamiento asintótico de las soluciones que cambian de signo en tres casos diferentes: cuando la parte positiva y negativa de las soluciones no interactúan para ningún tiempo t ; cuando la parte positiva y negativa de la temperatura Γ(u) no interactúan para ningún tiempo t, y cuando la parte positiva y negativa de la temperatura Γ(u) interactúan pero el comportamiento de las soluciones viene dado por el del problema de Stefan de una fase después de cierto tiempo. xix

21 xx

22 Introduction Diffusion is the natural process by which, for example matter is transported from one part of a system to another as a result of random molecular motions. The classical experiment that illustrates this is the one in which a drop of ink is leaved in a vessel full of water, and it eventually spreads out around the container and all the whole solution appears uniformly coloured. (There exist more refined experiments to make sure no convection is present). Diffusion models appear in sciences as diverse as biology, thermodynamics, medicine, and even economics. In biology, population models study the population dynamics, i.e., shortterm and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. Population dynamics deals with the way populations are affected by birth and death rates, and by immigration and emigration. In medicine, the diffusion models are used to describe the growth of cancerous tumors, for example. In thermodynamics, the heat equation models the heat conduction, this is when an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature. In economics, the diffusion models fluctuations in the stock market, by using Brownian motion. There are two ways to introduce the notion of diffusion: either a phenomenological approach starting with Fick s laws of diffusion, or a physical and atomistic one, by considering the random walk of the diffusing particles. First, let us introduce the laws that rule the diffusion processes: The Fick s laws. Fick s first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. Then in 1-dimension we have F = D u x, (24) where F is the diffusion flux, u is the concentration of the diffusing substance, and D is the diffusion coefficient. On the other hand, from Fick s first law and the mass conservation in absence of any chemical reaction: u t + F =. (25) x Then form (24) and (25), we obtain Fick s second law that predicts how diffusion causes the xxi

23 concentration to change with time: u t = D 2 u x 2. (26) For the case of diffusion in two or more dimensions Fick s second law is given by u t = D u. (27) The local diffusion model is given by (27) plus some boundary and initial conditions which are needed to complete the model. From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein in 195. Another kind of diffusion models are the nonlocal diffusion models. These models can be derived from a variation of a position-jump process (see for example [35]). Consider a single specie in an N-dimensional habitat where it is presumed that the population can be adequately modeled by a single function u(x, t), which is the density at position x at time t. A continuous model for the population dynamics for species can be derived by considering in detail a situation discrete in both space and time, and then letting the size and time intervals become small. The classic derivation of the Laplacian, (27) via a random walk is given assuming a binomial distribution. We reproduce the derivation of the nonlocal model for the case N = 1. The habitat will be R. First, divide into contiguous sites, each of length x. Discretize time into steps of size t. Let u(i, t) be the density of individuals in site i at time t. We wish to derive the change in the number of individuals in this site during the next time interval. The first assumption is that the rate at which individuals are leaving site i and going to site j is constant. Thus the total number of individual leaving location i to location j should be proportional to: the population in the interval i, which is u(i, t) x; the size of the target site, which is x; and the amount of time during which the transit is being measured, t. Let J(j, i) be the proportionality constant. Then, the number of individuals leaving site i during the interval [t, t + t] is M J(j, i)u(i, t)( x) 2 t. (28) j= M j i During the same time interval, the number of arrivals to site i from elsewhere is M j= M j i J(i, j)u(j, t)( x) 2 t. (29) Combining (28) and (29), we deduce that the populations density at location i and time t+ t xxii

24 is given by u(i, t + t) x = u(i, t) x + M j= M j i then, dividing (3) by x, we obtain u(i, t + t) = u(i, t) + M j= M j i J(i, j)u(j, t)( x) 2 t J(i, j)u(j, t) x t M j= M j i M j= M j i J(j, i)u(i, t)( x) 2 t, (3) J(j, i)u(i, t) x t. (31) Thus, allowing t and x in (31), we obtain ( ) u t (x, t) = J(x, y)u(y, t) J(y, x)u(x, t) dy. (32) Now, let us reinterpret equation (32), with R. We assume J(x, y) is a positive function defined in that represents the density of probability of jumping from a location y to x, and u(x, t) is the density of population at the point x at time t, then J(x, y)u(y, t)dy is the rate at which the individuals arrive to location x from all other locations y. Since we have assumed that J is the density of probability, and J is defined in, then J(x, y)dy = 1. In particular, u(x, t) = J(x, y)dy u(x, t) is the rate at which the individuals are leaving from location x to all other locations y. Then, we can write the equation (32) with initial condition u, as u t (x, t) = J(x, y)u(y, t)dy u(x, t), x, (33) u(x, ) = u (x), x. This problem and variations of it have been previously used to model diffusion processes, in [2], [18], [27], and [35], for example. This model allows to take into account short-range and long-range interactions, and it is possible to generalize the problem (33), for R N, or even more general type of measurable set, (see Chapter 1). The model (33) is called nonlocal diffusion model since the diffusion of the density u at point x and time t does not only depend on u(x, t), but on all the values of u in a neighbourhood of x through the convolution term J(x, y)u(y, t)dy. Now, let us fix an open set R N. For local problems as (27) the two most usual boundary conditions are Neumann s and Dirichlet s. The local heat equation with Neumann boundary condition is given by u t (x, t) = u(x, t), x, t >, u (x, t) =, x, t >, (34) ν u(x, ) = u (x), x, where ν denotes the (typically exterior) normal to the boundary, and u ν = models that the individuals do not enter or leave. An analogous nonlocal problem defined in R N xxiii

25 open, proposed in [18], is given by u t (x, t) = J(x, y) (u(y, t) u(x, t)) dy = u(x, ) = u (x), J(x, y)u(y, t)dy h (x)u(x, t) (35) where J : R N R N R, with R J(x, y)dy = 1, and we denote N h (x) = J(x, y)dy, x. In (35), the integral is over, then this model assumes that individuals may not enter or leave, and the diffusion takes place only in. Moreover, (35) shares with the local problem (34), that the constants are equilibrium solutions. On the other hand, the local heat equation with homogeneous Dirichlet boundary conditions is given by u t (x, t) = u(x, t), x, t >, u(x, t) =, x, t >, u(x, ) = u (x), x. In this case, u is zero in the boundary of the habitat. An analogous nonlocal problem proposed in [18] with R N open, is given by u t (x, t) = J(x, y)u(y, t)dy u(x, t), x, t >, R N (36) u(x, t) =, x /, t >, u(x, ) = u (x), x. In this model the diffusion takes place in the whole R N, and u = outside. Hence, this problem models the case in which the individuals extinguish when they leave the habitat, and R J(x, y)u(y, t)dy = N J(x, y)u(y, t)dy. Therefore, the equation in (36), is given by u t (x, t) = J(x, y)u(y, t)dy u(x, t). Problems (32), (33), (35) and (36) can be unified considering the nonlocal problem u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t), x, t >, u(x, ) = u (x), x, with h defined in. This is the kind of linear nonlocal problems we are going to study. Concerning nonlinear problems, we introduce the nonlocal reaction-diffusion model, by adding a local reaction term f(x, u(x, t)) to the diffusion population model (37), u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t) + f(x, u(x, t)), x, t >, (38) u(x, ) = u (x), x, xxiv (37)

26 where f : R R. This model was also considered in [35], where (38) models the population dynamics of species, and f denotes the per capita net reproduction rate at x at the given population density u(x, t), to take into account the number of new individual at x at time t. We also consider nonlocal reaction-diffusion models, with nonlocal diffusion and nonlocal reaction. The problem is given by u t (x, t) = J(x, y)u(y, t)dy h(x)u(x, t) + f(x, u)(, t), x, t >, (39) u(x, ) = u (x), x, but now f : L 1 () R is a nonlocal term. This model has been previously considered in [3]. The problem with local diffusion, ( ), and nonlocal reaction has been considered in [11], where the nonlocal reaction term takes into account a nonlocal saturation, or nonlocal competition effects. Another kind of nonlocal diffusion model, is the one that appears in [12, 19], given by u t (x, t) = J(x, y)(γ(u(y, t)) Γ(u(x, t)))dy, x R N, t >, (4) R N where Γ(u) = sign(u) ( u 1 ). This problem is called the nonlocal Stefan problem, which + models the temperature and enthalpy distribution in a phase transition between several states, for example the phase change from ice to water. Recently, there has been a big interest in studying diffusion in spaces which are non smooth. There are many attempts to try to generalize the laplacian to nonsmooth spaces. There are Dirichlet forms, that help describing jump processes which can be defined in spaces that are nonsmooth. Hence, it is possible to define differential equation on nonsmooth spaces, like some fractal sets. With this theory, called Analysis at fractals, it is possible to extend concepts like the laplacian, Green s functions and heat kernels, (see [9, 37, 5]). Nonlocal diffusion models like (37), (38), (39) can be naturally defined in metric measure spaces (see Chapter 1), since we just need to consider the density of probability of jumping from a location x in to a location y in, given by the function J(x, y). And this kind of density can be defined in a general metric measure space, since, we just need the space to have a measure and a metric. This allows us studying the diffusion in very different type of spaces, like: graphs, (which are used to model complicated structures in chemistry, molecular biology or electronics, or they can also represent basic electric circuits into digital computers), compact manifolds, multi-structures composed by several compact sets with different dimensions, (for example a dumbbell domain, where it is necessary to consider a perturbed domain to study local diffusion problems, as we can see in [3], whereas in the nonlocal diffusion problems we will be able study the problem directly in the domain), or even fractal sets as the Sierpinski gasket. xxv

27 Let us focus in what will be done throughout this work. As we said above, in this thesis we study general nonlocal diffusion problems. Let µ be a measure and d a metric defined in, we consider (, µ, d) a metric measure space, which is introduced in Chapter 1. First of all, we consider the linear nonlocal diffusion problem which is given by { u t (x, t) = (K hi)(u)(x, t), x, t > (41) u(x, ) = u (x), x, where is the integral operator, and K(u)(x, t) = J(x, y)u(y, t)dy hi(u)(x, t) = h(x)u(x, t) is the multiplication operator with h L () or in C b (), where C b () are the continuous and bounded functions defined on. We will not assume, unless otherwise made explicit, that J(x, y)dy = 1. A function that will be important throughout this work is h (x) = J(x, y)dy, which is not necessarily equal to the identity. To study the linear problem (41), in Chapter 2, we first derive a complete study of the linear operator K hi, studying the spaces where the operators are defined, the compactness and the spectrum of K and hi separately. Then in Chapter 3 we concentrate on the study of existence and uniqueness of the solution of (41), as well as the monotonicity properties of the solution in X = L p (), with 1 p or X = C b (). We recover and generalize the study of existence and uniqueness of solution of (41), with h = h or h = Id, and R N a domain, has been previously done in [2, 18] in L 1 (). After this, we study in detail the asymptotic behaviour of the solution as time goes to infinity. We prove that if σ X (K hi) is a disjoint union of two closed subsets σ 1 and σ 2 with Re(σ 1 ) δ 1, Re(σ 2 ) δ 2, with δ 2 < δ 1, then the asymptotic behavior of the solution of (3.1) in X is described by the Riesz Projection of K hi corresponding to σ 1. We prove also that the Riesz projection and the Hilbert projection are equal. Furthermore, we apply this result to the particular cases of the nonlocal diffusion problem (3.1) with h constant or h = h. In particular, we recover and generalize the result in [18], for X = L p (), with 1 p or X = C b (), whereas in [18], the authors obtain the result with R N an open set, in L 2 () if the initial data is in L 2 (), and in L () if the initial data is in C(). The study of the problem (41) leads us to the conclusion that equation (41) shares some properties with the classical heat equation, in particular, they both have weak and strong maximum principles, when J satisfies hypotheses of positivity, but they do not share the regularizing effect, as was pointed in [27], in the case = R N. This happens because the solution of (41) carries the singularities of the initial data. However, we have been able to prove that the semigroup S(t) of (41) satisfies that S(t) = S 1 (t) + S 2 (t), with S 1 (t) that converges to as t goes to infinity in norm X, and S 2 (t) is compact, hence S(t) is asymptotically smooth, xxvi

28 according to the definition in [32, p. 4]. In Chapter 4, we consider a nonlocal reaction-diffusion equation, with a nonlinear reaction term, and we work with the problem { u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)), x, t > u(x, t ) = u (x), x, (42) with f : R R, and initial data u L p (). The function f(x, s) will be assumed to be locally Lipschitz in the variable s R, uniformly with respect to x. There exists a large literature in the study of the local nonlinear reaction- diffusion equation { u t (x, t) = u(x, t) + f(x, u(x, t)), x, t >, (43) u(x, t ) = u (x), x. Existence, uniqueness and comparison results of the solutions of (43) with nonlinear locally Lipschitz term, f, as in (42) satisfying sign conditions are well-known, see for example [47, 4]. The arguments used for the problem (43) are essentially fixed-point arguments, but we can not use these arguments for the nonlocal problem (42), because the linear semigroup S(t) associated to (41) does not regularize. We prove first the existence for the equation (42) with f globally Lipschitz, and secondly, we prove the existence for f locally Lipschitz satisfying sign conditions with sub-supersolution arguments, in X = L p (), with 1 p or X = C b (). Hence, we recover and generalize the results of existence and uniqueness of solutions of (42), with R N and initial data in C(), in [8]. Observe that in [3], the authors study Fujita exponents for (42), which coincides with the classical one, (43). We will also study the asymptotic behaviour of the solution of (42). In [44], under sign conditions on the nonlinear term, the authors prove the existence of two extremal equilibria of (43), with R N a bounded domain and different type of boundary conditions. The authors also prove that the asymptotic dynamics of the solutions enter between these extremal equilibria, uniformly in space, for bounded sets of initial data. As a consequence, they obtained a bound for the global attractor for the local reaction-diffusion equations. On the other hand, we prove that there exist also two ordered extremal equilibria ϕ m and ϕ M (one minimal and another maximal), for the problem (42), and all the asymptotic dynamics of the solutions of (42) with bounded initial data, enter between the two extremal equilibria ϕ m and ϕ M, when time goes to infinity in X = L p (), with 1 p <. Moreover, the same extremal equilibria, ϕ m and ϕ M, are bounds of any weak limit in L p (), with 1 p <, of the solution of (42) with initial data u in L p (). Observe that for the nonlocal problem (42), we obtain a weaker result than for the local problem (43) again by the lack of smoothing effects. After studying the asymptotic behaviour, we discuss the existence and stability of equixxvii

29 librium solutions for the problem u t (x, t) = J(x, y)u(y, t)dy h (x) u(x, t) + f(u(x, t)), x, t > u(x, ) = u (x), x, (44) Let F be the Nemitcky operator associated to the function f, such that F (u)(x, t) = f(u(x, t)). Since F : L p () L p () is not differentiable (see Appendix B), and the semigroup associated to the linear problem (41) does not regularize then the principle of linearized stability fails. However, under hypotheses on the convexity of the function f, we prove that the stability/instability with respect to the linearization, implies the stability/instability of the equilibria of the nonlinear problem (44). We will also prove that any continuous nonconstant equilibrium solution of (44) is, if it exists, unstable when f is convex. In [16], [14] and [4], the authors prove similar results for the local reaction-diffusion problem (43) with Neumann boundary conditions. In [14] and [4], the authors also prove that if is a convex domain, then any nonconstant equilibrium, is, if it exists, unstable for any dimension. Up to our knowledge this result has not been proved for the nonlocal problem (44), and the techniques used for the local problem do not seem to be useful to prove the instability of nonconstant equilibria if the domain is convex. There exists a big interest in the study of the existence and stability of equilibria of the problem (42). In [8], the authors study the stability of the positive steady solutions, with initial data in C(). In particular they prove, under hypothesis on the spectrum of the linear operator K J, that there exists a unique nonnegative equilibrium asymptotically stable in C() +. In Chapter 5, we study the nonlocal reaction-diffusion problem with both terms nonlocal, i.e., we consider the problem { u t (x, t) = (K hi)(u)(x, t) + f(x, u)(, t), x, t > (45) u(x, ) = u (x), x, where (K hi)(u) is the nonlocal diffusion term and f : L 1 () R is the nonlocal reaction term, and it is defined as f = g m, where g : R R is a nonlinear function, and m : L 1 () R is the average of u in a ball of radius δ > and center x, defined as 1 m(x, u(, t)) = u(y, t)dy. µ(b δ (x)) B δ (x) We first derive a complete theory of existence and uniqueness for the problem (45), in X = L p (), with 1 p or X = C b (), with g globally Lipschitz. The problem (45) with g linear, may fail to have comparison properties. Hence, we give comparison results for (45), with g globally Lipschitz, with Lipschitz constant small enough, using fixed-point arguments. xxviii

30 If g is locally Lipschitz, and satisfies sign conditions then we will prove the existence and uniqueness of solution of the problem (45), with nonlinear term g, such that the Lipschitz constant of g k is small enough, where g k is a truncated function associated to g. In fact, the existence and uniqueness, will be proved for initial data in L (), such that u L () k. Furthermore, we will prove some monotonocity properties for the solution of (45) with g and u satisfying the conditions above. We prove also that all the asymptotic dynamics of the solutions of (45) with g globally Lipschitz, enters between two extremal equilibria ϕ m and ϕ M, as we do for the nonlocal reaction-diffusion problem (42). In fact, the asymptotic dynamics of the solutions of (45) enters between ϕ m and ϕ M uniformly in compact sets of. Another advantage of this model (45), with nonlocal reaction with respect to the nonlocal reaction-diffusion problem (42), is that the nonlocal reaction term f : L p () L 1 () is compact, and we prove that the semigroup associated to (45) is asymptotically smooth, and then we use [32, Theorem ] to prove the existence of a global attractor for the semigroup of (45). In Chapter 6, we study the nonlocal two-phase Stefan problem in R N u t = J(x y)v(y)dy v, where v = Γ(u), R N u(, ) = f, (46) where J is a smooth nonnegative convolution kernel, u is the enthalpy and Γ(u) = sign(u) ( u 1 ) +. The Stefan problem is a non-linear and moving boundary problem which aims to describe the temperature and enthalpy distribution in a phase transition between several states. The history of the problem goes back to Lamé and Clapeyron [39], and afterwards [49]. For the local model can be seen e.g. the monographs [17] and [54] for the phenomenology and modeling; [23], [41], [45] and [53] for the mathematical aspects of the model. The main model uses a local equation under the form u t = v, v = Γ(u), but recently, a nonlocal version of the one-phase Stefan problem was introduced in [12], which is equivalent to (46) in the case of nonnegative solutions, and Γ(u) is given by Γ(u) = ( u 1 ) +. This new mathematical model turns out to be rather interesting from the physical point of view at an intermediate (mesoscopic) scale, since it explains for instance the formation and evolution of mushy regions (regions which are in an intermediate state between water and ice). We study the existence, uniqueness and comparison results along the lines of the previous Chapters, and we study the asymptotic behaviour in the spirit of [12], but for sign-changing solutions, which presents very challenging difficulties concerning the asymptotic behavior. Though we do not give a complete study of the question which appears to be rather difficult, we give some sufficient conditions which guarantee the identification of the limit when time goes to infinity. xxix

31 Below, we briefly summarize the organization of the work: In chapter 1 we describe the metric measure spaces (, µ, d), and enumerate the nonlocal diffusion models that will be studied in the following chapters. In chapter 2 we study the linear operator ( K hi ) (u)(x) = J(x, y)u(y)dy h(x)u(x), x. We start studying the operator K(u), which has a straight dependence with the kernel J. We give results of regularity and compactness of the operator K, in terms of the regularity of J. We study the positiveness of the operator K, and we describe the spectrum of the operator K. We will give conditions to obtain that the spectrum is independent of the Lebesgue space where we are working. After that we study the multiplication operator hi, that sends u(x) to h(x)u(x). In the last part of this chapter we analyze the spectrum of K hi, and we will also give conditions to obtain that the spectrum of K hi is independent of the Lebesgue space. In chapter 3 we give a result of existence and uniqueness of solution of (41). We write the solution in terms of the group associated to the operator K hi. We give also monotonicity results. We prove that under some hypothesis on the positivity of the kernel J, the Weak and Strong Maximum Principle. In the last part of this chapter, we prove that the solutions of the homogeneous problem (36) converges asymptotically to the eigenfunction associated to the first eigenvalue of the operator K hi, and the solution of the problem (35) has an exponential convergence to the mean value of the initial data u L p (), with 1 p. In chapter 4 we work with the nonlinear problem (42). We give a result of existence and uniqueness of solutions for f locally Lipschitz satisfying an increasing property. We also prove the existence of two extremal equilibria solution (one maximal and another minimal). We prove that all the solutions enter between these two extremal equilibria when time goes to infinity. We study also in the particular case of the problem (44) that if the reaction term f is strictly convex, any nonconstant equilibrium solution is unstable, if it exists. In chapter 5 we are confined to the nonlocal reaction-diffusion problem (45), with both terms nonlocal. We give a result of existence and uniqueness and comparison results of solutions of the problem (45), with g globally lipschitz. We give also a result of existence and uniqueness of solution of the problem (45) with g locally Lipschitz and sublinear and some bounded initial data. We prove that the asymptotic dynamic of the solutions enter between two extremal equilibria when time goes to infinity, and finish proving the existence of a global attractor. In chapter 6, we study the nonlocal two-phase Stefan problem in R N. We give results of existence and uniqueness of solutions of (46). And we focus in the study of the asymptotic xxx

32 behaviour of sign-changing solutions. Since, we have not been able to give a general result about this asymptotic behavior, we give some sufficient conditions to guarantee the identification of the limits. This work has been done in collaboration with Professor Emmanuel Chasseigne at the Université François Rabelais in Tours, during my stay for three months in 211. xxxi

33 xxxii

34 Chapter 1 Nonlocal diffusion on metric measure spaces First of all we introduce some Measure Theory, to define the metric measure spaces, [46]. Then we enumerate some examples of metric measure space, in which all the theory throughout this work can be applied, we consider open subsets of R N, which are the most usual in the literature; graphs, which have plenty of applications; compact manifolds; multi-structures, that are the union of metric measure spaces of different dimensions, etc. We finish introducing the linear nonlocal diffusion model, and we enumerate the different problems that will be analyzed in the following chapters. 1.1 Metric measure spaces In this section we introduce concepts of Measure Theory, for more information see [46]. First of all, let us start defining what is a metric space (X, d) that consists of a set X and a distance d on X, i.e., a function d : X X [, ) satisfying the following properties: for all x, y, z X i. d(x, y) and d(x, y) = if and only if x = y, ii. d(x, y) = d(y, x), iii. d(x, y) d(x, z) + d(z, y). We will denote the balls in X by B(x, r) = {y X : d(x, y) < r} where x X and r >. Let us introduce now several definitions. Definition i. A collection M of subsets of X is said to be a σ-algebra in X if M has the following properties: (a) X M. 1

35 (b) If A M, then A c M, where A c = X \ A. (c) If A = A n, and A n M for n = 1, 2,..., then A M. n=1 ii. Let X be a topological space, we denote by B the smallest σ-algebra in X such that every open set in X belongs to B. The members of B are called the Borel sets of X. Definition i. A positive measure is a function µ, defined on a σ-algebra M, whose range is in [, ] and which is countably additive. This means that if {A n } n N is a pair-wise disjoint collection of members of M, then ( ) µ A n = µ(a n ). n=1 To avoid trivialities, we shall also assume that µ(a) < for at least one A M, A. ii. A measure space, (X, M, µ), has a positive measure defined on the σ-algebra, M, and the members of M are called the measurable sets in X. iii. Let X be a measure space, Y be a topological space, and f : X Y, then f is said to be measurable provided that f 1 (V ) is a measurable set in X for every open set V in Y. iv. A measure µ is called complete measure if every subset of a set with measure zero is measurable. v. The measure µ defined on a σ-algebra M in X is σ-finite measure if X is a countable union of sets X i with finite measure. vi. We call total variation of µ to the function µ defined on the Borel σ-algebra B in X by µ (E) = sup µ(e i ), E B i=1 n=1 the supremum being taken over all disjoint partitions {E i } of E. The following Theorem states that every measure can be completed (see [46, p. 28]). Theorem Let (X, M, µ) be a measure space, let M be the collection of all E X for which there exists sets A and B in M such that A E B and µ(b A) =, and define µ(e) = µ(a) in this situation. Then M is a σ-algebra, and µ is a measure on M. Thanks to this result, whenever it is convenient, we may assume that any given measure is complete. The following result can be found in [46, p. 4]. 2

36 Theorem Let X be a locally compact Hausdorff space. Then there exists a σ-algebra M in X which contains all Borel sets in X, and there exists a positive measure µ on M such that i. µ(k) < for every compact set K X. ii. For every µ-measurable set E M we have that iii. The relation µ(e) = inf{µ(v ) : E V, V open}. µ(e) = sup{µ(k) : K E, K compact}. holds for every open set E, and for every E M, with µ(e) <. iv. If E M with µ(e) = and A E, then A M. (µ is complete). A measure µ defined on the σ-algebra of all Borel sets in a locally compact Hausdorff space X is called a Borel measure on X. If µ is positive, a Borel set E X is outer regular or inner regular, if E has property ii. or iii., respectively, of Theorem If every Borel set in X is both outer and inner regular, µ is regular. Throughout this work we will be working with metric measure spaces, and any time we mention them, we will be referring to the following definition. Definition A metric measure space (X, µ, d) is a metric space (X, d) with a σ- finite, regular, and complete Borel measure µ in X, that associates a finite positive measure to the balls of X. Remark Let (X, µ, d) be a metric measure space, according to the previous definition, the measure satisfies the properties in Theorem The measure µ is a complete and regular measure, which are the properties ii., iii. and iv. in Theorem 1.1.4, and moreover, since µ associates a finite positive measure to the balls of X, then for every compact set K X, µ(k) <, then the property i. in Theorem is satisfied Function spaces in a metric measure space Let (, µ) be a measure space where µ is a measure as in Definition For 1 p <, if f is a measurable function on, we define ( f L p () = f p dµ and let L p () consist of all f for which f L p () <. We call f L p () the L p -norm of f. Let f be a measurable function on. The essential supremum of f : R, ess sup(f), is defined by ess sup(f) = inf{a R : µ ( {x : f(x) > a} ) = }. 3 ) 1/p

37 For p =, if f is a measurable function on, we define f L () to be the essential supremum of f, and we let L () consist of all f for which f L () <. In particular, if µ() < and q > p then L q () L p (). Let (, µ, d) be a metric measure space, if f is a measurable function on, we define f Cb () by the supremum of f, and we let C b () consist of all continuous and bounded functions f, such that f Cb () <. Then C b () L (). The results that will be used throughout this work, and are well known properties of the L p -space are: Hölder s inequality and Minkowski s inequality; the Monotone Convergence Theorem and the Dominated Convergence Theorem; Fubini s Theorem and Lusin s Theorem. (These results can be see in detail in Appendix A). Furthermore, since L p (), with 1 p < is a Banach space, we can consider its dual which is given by L p (), for p satisfying 1/p + 1/p = 1, and the dual space of L () is (L ()) = M(), where M() is the set of measures satisfying the properties in Theorem Some examples of metric measure spaces In the following chapters we will consider a general measure metric space (, µ, d). In this section we enumerate some examples to which we can apply the theory developed throughout this work. Subset of R N : Let be a Lebesgue measurable set of R N with positive measure. A particular case is the one in which is an open subset of R N, which can be even = R N. We consider the metric measure space (, µ, d) with: R N, µ the Lebesgue measure on R N, d the Euclidean metric of R N. Graphs: We consider a graph G = (V, E), where V R N is the finite set of vertices, and the edge set E, consists of a collection of Jordan curves E = { π j : [, 1] R N j {1, 2, 3,..., n} } where π j C 1 ([, 1]) is injective. We consider that each e j := π j ([, 1]) has its end points in the set of vertices V, and any two edges e j e h satisfy that the intersection e j e h is either empty, 1 vertex or 2 vertices. We consider a graph in R N, non empty, connected and finite. identify the graph G R N with its associated network. n n G = e i = π j ([, 1]). j=1 4 j=1 From now on, we

38 We denote v = π j (t) for some t [, 1]. For a function u : G R we set u j := u π j : [, 1] R, and use the abbreviation ( u j (v) := u j π 1 j (v) ) We define the measure structure of this graph. The edges have associated the Lebesgue measure in dimension 1, and the length of the edge e i is defined as the length of the curve π i, µ(e i ) = µ(π i [, 1]) = 1 π i(t) dt. (1.1) A set A e i is measurable if and only if πi 1 (A) [, 1] is measurable, and for any measurable set A e i, we consider the measure µ i µ i (A) = π 1 i (A) π i(t) dt. Hence a set A G is measurable if and only if A e i is measurable for every i {1, 2, 3,..., n}, and its measure is given by µ(a) = n µ i (A e i ). (1.2) i=1 It turns out that a function f : G R is measurable if and only if f ei : e i R is measurable. n For 1 p <, we set f L p (G) = L p (e i ), with norm where, i=1 f L p (G) = n f L p (e i ) <, i=1 ( 1 1/p ( 1 ) 1/p f L p (e i ) = f(π i (t)) p π i(t) dt) = f(π i ( )) p dµ i. For p =, f L (G) = where, n L (e i ), with norm i=1 f L (G) = max f L i=1,...,n (e i ) <, f L (e i ) = sup f(π i (t)). t [,1] Furthermore, a function f : G R is continuous in the graph G, if and only if f ei : n e i R is continuous. We set f C(G) = C(e i ), with the norm associated f C(G) = i=1 max f C(e i=1,...,n i ) <, 5

39 where, f C(ei ) = sup f(π i (t)). t [,1] Now, let us describe the metric associated to the graph. For v, w G the geodesic distance from v to w is the length of the shortest path from v to w. This distance will be the metric associated to the graph G, and we denote the geodesic metric as d g. Moreover, since the graph is connected, there always exists the path from v to w, and since the graph is finite the geodesic metric d g is equivalent to euclidean metric in R N. Let us see this below: The graph G is compact in (G, d g ), the graph with the geodesic metric, and G is compact in (G, d), the graph with the euclidean metric in R N. We consider the identity map I : (G, d g ) (G, d), thus, we have that I is continuous, because for any v, w G with d(v, w) d g (x, y). Thus, since I is continuous and injective in a compact set, and Im(G) = G, then I is an homeomorfism. Therefore, the metrics d g and d are equivalent. To sum up, the metric measure space (G, µ G, d g ) is given by: G is a graph with a finite number of edges and vertices, µ G the measure described in (1.2), d g is the geodesic metric which is equivalent to the Euclidean metric of R N Manifolds, Multi-structures and other metric measure spaces Let us introduce the family of Hausdorff measures below, for which we follow [26, chap. 2]. A d-dimensional Hausdorff measure is a type of positive outer measure, that assigns a number in [, ] to a set in R N. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or if the set is infinite. The one-dimensional Hausdorff measure of a simple curve in R N is equal to the length of the curve. Likewise, the two-dimensional Hausdorff measure of a measurable subset of R 2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, area and volume. In fact, there are d-dimensional Hausdorff measures for any d, which is not necessarily an integer. Definition i. Let (R n, d) be the euclidean metric space. For any subset E, let diam(e) denote its diameter, diam(e) = sup{d(x, y) : x, y E}, diam( ) =. Let E be any subset of, and δ > a real number. We define { } Hδ s (E) = inf (diam(e i )) s : E E i, diam(e i ) < δ. i=1 6 i=1

40 ii. For E and s as above, we define H s (E) = lim Hδ s (E) = sup Hδ s (E). δ δ> We call H s the s-dimensional Hausdorff measure on R n. In the following result, we give several properties of the Hausdorff measure. Theorem (Elementary properties of Hausdorff measure) i. H s is a Borel regular measure in R N for s <. ii. H is a counting measure. iii. H N is the Lebesgue measure in R N. iv. H s on R N for all s > N. v. Let A R N and s < t <. (a) If H s (A) <, then H t (A) =. (b) If H t (A) >, then H s (A) = +. We define below the Hausdorff dimension of a subset of R N. Definition The Hausdorff dimension of a set A R N is defined to be H dim (A) inf{ s < : H s (A) = } Remark Observe H dim (A) N. If we denote s = H dim (A), then H t (A) = for all t > s and H t (A) = + for all t < s; H s (A) may be any number between and included. Furthermore H dim (A) need not be a integer. Even if H dim (A) = k is an integer and < H k (A) <, A need not be a k-dimensional surface in any sense. Let us introduce more examples of metric measure spaces. Compact Manifold: Let M R N be a compact manifold that we define as follows: Let U be an open bounded set of R d, with d N, and let ϕ : U R N be an application such that it defines a diffeomorphism from U onto its image ϕ(u), then we define the compact manifold as M = ϕ(u). A natural measure in M, is the one for which, A M is measurable if and only if ϕ 1 (A) R d is measurable. Hence for any measurable set A M, we define the measure µ as, (see [48, p. 48]) µ(a) = g dx, (1.3) where g = det(g ij ) and g ij = ϕ x i, ϕ x j. 7 ϕ 1 (A)

41 Since the compact manifold M R N is given by ϕ(u), with U R d, then the ambient measure is the d-dimensional Hausdorff measure for the manifold M. In fact, the measure (1.3) is equal to the d-hausdorff measure, (see [48, p. 48]). The natural metric in M: Let l(c) be the length of the curve defined as in (1.1), then we define the geodesic distance between two points p, q in a manifold M as, (see [29, p. 164]): d g (p, q) := inf{l(c) c : [, 1] M smooth curve, c() = p, c(1) = q}. (1.4) On the other hand M R N then the ambient metric is the euclidean metric, d. Since the manifold M is compact, and arguing like we did for the graph G, we obtain the the geodesic metric, d g, and the euclidean metric, d, are equivalent. To sum up, the metric measure space (M, H d, d) is given by: M the compact manifold in R N. H d the d-dimensional Hausdorff measure. d g is the geodesic metric equivalent to the Euclidean metric of R N. Multi-structure: Now, we consider a multi-structure, composed by several compact sets with different dimensions. For example, we can think in a piece of plane joined to a curve that is joined to a sphere in R N, or we can think also in a dumbbell domain. Therefore, we are going to define an appropriate measure and metric for these multistructures. Let (X, µ X, d) be the direct sum of metric measure spaces composed by a collection of metric measure spaces { (X i, µ i, d i ) } i {1,...,n}, with its respective measures, µ i, and metrics, d i, defined as above. Moreover, we assume the measure spaces { (X i, µ i ) } i {1,...,n} satisfy µ i (X i X j ) = µ j (X i X j ) =, for i j, and i, j {1,..., n}. We define X = i {1,...,n} X i, (1.5) and we say that E X is measurable if and only if E X i is measurable for all i {1,..., n}. Moreover we define the measure µ X as µ X (E) = n µ i (E X i ). (1.6) i=1 Furthermore, let us define the metric that we consider in X. We assume that X i R N is compact for all i {1,..., n}, and the metrics d i associated to each X i, are equivalent to the euclidean metric in R N. Therefore, the metric d that we consider for the multistructure, is the euclidean metric in R N. To sum up, the metric measure space (X, µ X, d) is given by: 8

42 X the multi-structure in (1.5). µ X the measure given by (1.6). d is the Euclidean metric of R N. Space with finite Hausdorff measure and geodesic distance: We consider a compact set F R N, with H dim (F ) = s, and such that F has finite s-hausdorff measure, i.e., H s (F ) <. The metric associated to F is the geodesic metric, which may not be equivalent to the euclidean metric in R N. Therefore, we consider the metric measure space (F, µ F, d g ) given by: F is a compact set in R N. H s the s-dimensional Hausdorff measure. d g is the geodesic metric. There exist some examples with the previous metric and measure associated, some are fractal sets like the Sierpinski gasket, (see Figure 1.1). The Sierpinski gasket is a fractal set that has associated a metric and measure like the ones described above, i.e., we consider the log(3) log(2) - dimensional Hausdorff measure Hs, and the geodesic metric. For more information see [37] and [2]. Figure 1.1: Sierpinski Gasket. 1.3 Nonlocal diffusion problems Now, let us introduce the kind of linear nonlocal diffusion problems we are going to deal with throughout this work. We describe first the problem in a general metric measure space (, µ, d), or in subsets of R N : - Diffusion in a metric measure space: Let (, µ, d) be a metric measure space, and let u(x, t) be the density of population at the point x at time t. We assume J is a positive function defined in, i,e., (x, y) J(x, y) and we assume that J is the density of probability of jumping from a location y to x, and u(x) is the density 9

43 of population at the point x, then J(x, y)u(y)dy is the rate at which the individuals arrive to location x from all other locations y. Since we have assumed that J is the density of probability, and J is defined in, then J(x, y)dy = 1, for all x. In particular, u(x) = J(x, y)dy u(x) is the rate at which the individuals are leaving from location x to all other locations y. Then, we consider the problem u t (x, t) = J(x, y) ( u(y, t) u(x, t) ) dy = J(x, y)u(y, t)dy u(x, t), x, t > u(x, ) = u (x), x. (1.7) In this problem, the integral terms only take into account the diffusion inside. Thus the individuals may not enter or leave. In particular, when R N, the diffusion is forced to act only in with no interchange of mass between and the exterior R N \. - Diffusion in R N : Let R N and let us assume that J(x, y) is the density of probability of jumping from x to y defined in R N R N, then we have that R N J(x, y)dy = 1 for all x R N. Therefore J(x, ) L 1 () = 1 and J(x, ) L 1 () for all x. Some examples of J s in R N are the following: J(x, y) = e x y 2 σ 2, where σ > (Normal distribution); J(x, y) = 1 x y α 1, where < α < 1. We are interested in two kind of nonlocal problems, which appear in [18]: The nonlocal problem proposed in [18] as an analogous problem to the local diffusion problem with Dirichlet boundary conditions, is the following: it is imposed u = g outside. Hence the nonlocal problem is given by u t (x, t) = J(x, y)u(y, t)dy u(x, t), x, t > R N (1.8) u(x, t) = g(x), x /, t > u(x, ) = u (x), x. Let us consider a nonlocal diffusion problem where the diffusion is forced to act only in R N, then the integrals over the whole R N that appear in (1.8) are replaced by integrals only in. The nonlocal diffusion problem (1.9) proposed in [18] as the nonlocal problem analogous to the classical heat equation with Neumann boundary conditions is given by u t (x, t) = J(x, y) (u(y, t) u(x, t)) dy, x, t > (1.9) u(x, ) = u (x), x. Now, we unify the nonlocal problems (1.8) and (1.9). 1

44 The problem (1.8) can be rewritten as: u t (x, t) = J(x, y)u(y, t)dy u(x, t) + where and = (K I)u(x, t) + G g (x) K(u)(x, t) = G g (x) = R N \ R N \ J(x, y)u(y, t)dy J(x, y)u(y, t)dy (1.1) J(x, y)g(y)dy. The problem (1.9) can be written as: u t (x, t) = J(x, y)u(y, t)dy J(x, y)dy u(x, t) = (K h I)u with K as in (1.1), and h (x) = J(x, y)dy. In particular, since J is nonnegative and J(x, y)dy = 1, for all x, we have that h (x) 1. R N with and We unify the nonlocal problems (1.8) and (1.9) as follows: { u t (x, t) = (K hi)(u)(x, t) + G g (x), x, t > u(x, ) = u (x), x, 1, for the problem (1.8), h(x) = h (x) = J(x, y)dy, for the problem (1.9), G g (x) = { G g (x), for the problem (1.8),, for the problem (1.9). (1.11) We consider now a metric measure space (, µ, d), which can be even R N, and we unify the problems (1.7), (1.8) and (1.9). Hence the problem we work with through this work is the following { u t (x, t) = (K hi)(u)(x, t), x, t > (1.12) u(x, ) = u (x), x, with h L () and K(u) = J(x, y)u(y)dy, where J : R. As we can see in (1.12), the problem is defined for x, and the integral operator K(u) acts only in. Now, we enumerate the equations we are going to work with throughout this work. Let (, µ, d) be a metric measure space: 11

45 i. In Chapter 3, we study the evolution linear nonlocal problem u t (x, t) = (K hi)(u)(x, t), x. (1.13) ii. In Chapter 4, we consider the the nonlocal reaction-diffusion equation. We add a local nonlinear reaction term f : R R to the equation (1.13). Thus, we study u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)), x. (1.14) iii. In Chapter 5, we consider a reaction-diffusion equation with a nonlocal reaction term, f : L 1 () R, with f = g m, where g : R R is a nonlinear function, and m is a average of u in the ball centered in x of radius δ. The reaction-diffusion equation is given by ( 1 u t (x, t) = (K hi)(u)(x, t) + g µ(b δ (x)) B δ (x) iv. In Chapter 6 we will study the two-phase Stefan problem in R N, u t (x, t) = J(x y)γ(u)(y, t)dy Γ(u)(x, t), R N where Γ(u) = sign(u)( u 1) +. Notation: Throughout this thesis we use the following notation: ) u(y, t)dy, x. (1.15) (, µ, d) will always be a metric measure space, with µ as in Definition Sometimes we will omit the part in which we mention µ as in Definition L p () is used to denote the Lebesgue spaces with p satisfying 1 = 1/p + 1/p, for 1 p. Let L p () be a Banach space. The dual space of L p (), will be considered as: for 1 p <, (L p ()) = L p (), where 1 = 1 p + 1 p, for p =, (L ()) = M(), where M() is the set of Radon measures, for more information see [28, chap. 7]. 12

46 Chapter 2 The linear nonlocal diffusion operator Throughout this chapter we will work with (, µ, d) a metric measure space, with the properties in Definition We consider the linear nonlocal diffusion problem: { u t (x, t) = (K hi)(u)(x, t), x, t >, u(x, ) = u (x), x (2.1) where (K hi)(u) = J(, y)u(y)dy h( )u, with J a function such that J : R, and h L (). In this chapter we give a comprehensive survey of the linear operator K hi, and in the next chapter we apply this theory to study the existence, uniqueness, positivity, regularizing effects and the asymptotic behavior of the solution of (2.1). We will start studying the linear nonlocal diffusive integral operator K(u) = J(, y)u(y)dy, (2.2) where J is the kernel of the operator. We will prove that under hypotheses on the integrability or continuity of J, K is a bounded linear operator in X = L p () or X = C b (). Moreover, under these same hypotheses on J, we will prove the compactness of the operator K. To prove the compactness, we will show that K can be approximated by operators with finite rank, and we will also use Ascoli-Arzela Theorem. We will denote the operator K by K J, to remark the dependence between J and K. We will also study the particular case of the convolution operator K J (u) = J ( y)u(y)dy, where J : R N R. 13

47 We give a result of positiveness of the diffusive operator K J : given a nonnegative function z, not identically zero, we will describe the set of points in where K J (z) is strictly positive. For this, we will assume that the kernel J satisfies J(x, y)> for all x, y, such that d(x, y)<r, (2.3) for some R > and R-connected (see Definition ). This positiveness will be used later on to prove that the solution of the problem (2.1) has a strong maximum principle. We are also interested in the adjoint operator associated to K J, which will be proved to be given by (K J ) = K J, where, J (x, y) = J(y, x). Moreover, if J satisfies that J(x, y) = J(y, x), then K J L(L 2 (), L 2 ()) is selfadjoint. In this case we have that the spectrum is real and it is bounded above and below by m = inf K J(u), u L u L 2, u L 2 =1 2 (),L 2 () and M = sup K J (u), u L 2 (),L 2 (). u L 2, u L 2 =1 Moreover, thanks to Kreĭn-Rutman Theorem, (see [38]), we will obtain that if the function J satisfies (2.3), then the spectral radius in C b () of the operator K J is a positive simple eigenvalue, with a strictly positive eigenfunction associated. A similar result was proved by Bates and Zhao [8], for R N open, but their hypothesis on the positivity of J is stronger, because they assume that J(x, y) > for all x, y. Let X = L p (), with 1 p or X = C b (). If K J L(L 1 (), C b ()) is compact, then we obtain that the spectrum σ X (K J ) is independent of X. Hence, the previous results will be also satisfied for the spectrum of K J in X. Therefore, σ X (K J ) [m, M] and the spectral radius of the operator K J in X will be proved to have a strictly positive associated eigenfunction. Finally, in the last part of this chapter we study the linear nonlocal operator K J hi, with h L () or h C b (). We will give Green s formulas for the operator K J hi when J(x, y) = J(y, x). A similar result can be found in [2], for R N open. Moreover, we will make a general spectral study of the operator K J hi, and we will prove that σ X (K J hi) is composed by Im(h), and eigenvalues of finite multiplicity. Furthermore, we will prove that if J(x, y) = J(y, x) and h h = J(, y)dy, then σ X(K J hi) is nonpositive. 2.1 Properties of the operator K We consider the function J, defined in as and define K J (u)(x) = x J(x, ) J(x, y)u(y)dy, with x for u defined in. We call J the kernel of the operator K J. We will not assume, unless otherwise made explicit, that has a finite measure. 14

48 2.1.1 Regularity of K J In this section we are going to study spaces between which the linear operator is defined, depending on the integrability or continuity of the function J. Moreover, we will prove that the operator is bounded. The following proposition states that under appropriate regularity of a general kernel J, we have that K J L(L p (), X), where X = L q (), C b () or X = W 1,q (), if R N is open. Proposition i. For 1 p, q, if J L q (, L p ()), then K J L(L p (), L q ()) and the mapping J K J is linear and continuous, and K J L(L p (),L q ()) J L q (,L p ()). (2.4) ii. For 1 p, if J L (, L p ()) and for any measurable set D satisfying µ(d) <, lim J(x, y)dy = J(x, y)dy, x, (2.5) x x D D then K J L(L p (), C b ()) and the mapping J K J is linear and continuous, and K J L(L p (),C b ()) J L (, L p ()). (2.6) In particular, if J C b (, L p ()), then K J L(L p (), C b ()), and K J L(L p (),C b ()) J Cb (, L p ()). iii. If R N is open, for 1 p, q, if J W 1,q (, L p ()), then K J L(L p (), W 1,q ()) and the mapping J K J is linear and continuous, and K J L(L p (),W 1,q ()) J W 1,q (,L p ()). (2.7) Proof. i. Thanks to Hölder s inequality, we have for 1 q < and 1 p, K J (u) q L q () = K J (u)(x) q dx q = J(x, y)u(y) dy dx u q L p () J(x, ) q dx = L p () u q L p () J q L q (,L p ()). For q = and 1 p, for each x, K J (u)(x) = J(x, y)u(y)dy u L p () J(x, ) L p (). (2.8) 15

49 Taking supremums in (2.8) in, we obtain K J (u) L () = sup K J (u)(x) u L p () sup J(x, ) L p () = u L p () J L (,L x x p ()). Thus, the result. ii. We have to prove, that for all u L p (), K J (u) C b (), for 1 p. The hypothesis (2.5) can also be written as lim J(x, y)χ D (y)dy = J(x, y)χ D (y)dy, x x x, (2.9) what means that K J (χ D ) is continuous in, where χ D is the characteristic function of D, with µ(d) <. Moreover, since J L (, L p ()), from i., we have that K J L(L p (), L ()), since µ(d) <, then χ D L p (), for 1 p. Then K J (χ D ) is bounded. Thus, K J (χ D ) C b () for any characteristic function χ D. Moreover, the space V = span [ χ D ; D with µ(d) < ], is dense in L p (), for 1 p. We prove it first for 1 p <. Suppose that there exists a function g (L p ()), such that g V, then, χ D (x)g(x)dx = g(x)dx =, D with µ(d) <, D what implies that g(x) = for a.e. x D, for all D with µ(d) <. On the other hand, µ is a σ-finite measure, then = i=1 D i, with µ(d i ) <. Thus, we have that g(x) = for a.e. x. Hence, V L p () densely, for all 1 p <. We prove it now for p =. We suppose that there exists a measure g (L ()) = M(), such that g V, where M() is the set of Radon measures (see Theorem 1.1.4), then χ D dg = g(d) =, D with µ(d) <, what implies that the measure g for all D with µ(d) <. Arguing like before, we obtain that V L () densely. From i., K J L(L p (), L ()), and we have already proved that K J : V C b (), and V L p () densely, then K J (L p ()) = K J ( V ) KJ (V ) C b (). Therefore, u L p (), with 1 p, K J (u) C b (). Finally, taking supremums in in (2.8), we obtain the result. In particular if J C b (, L p () ), then the hypothesis (2.5) is satisfied. Furthermore, C b (, L p () ) L (, L p () ). Thus, the result. 16

50 iii. As a consequence of Fubini s Theorem, and since is open we have that for all ϕ Cc () and i = 1,..., N, the weak derivative of K J (u) is given for all u L p () by Therefore xi K J (u), ϕ = K J (u), xi ϕ = J(x, y)u(y) xi ϕ(x) dy dx = J(x, y) xi ϕ(x)u(y) dx dy = J(, y), xi ϕ, u = xi J(, y), ϕ, u = xi J(x, y)ϕ(x)u(y) dx dy = xi J(x, y)u(y) ϕ(x) dy dx = K J x i (u), ϕ. (2.1) x i K J (u) = K J x i (u). (2.11) Since J W 1,q (, L p ()), and from part i. and (2.11), we have that for u L p () and i = 1,..., N, K J L(L p (),L q ()) J L q (,L p ()) (2.12) x i K J L(L p (),L q ()) = K x i J L(L p (),L q ()) J x i L q (,L p ()). (2.13) Hence, K J L(L p (), W 1,q ()), for all 1 p, q and from (2.12) and (2.13) we have (2.7). The following result collects the cases in which K J X = C b (). Corollary L(X, X), with X = L p () or i. If J L p (, L p ()) then K J L(L p (), L p ()), for 1 p fixed. ii. If J C b (, L 1 ()) then K J L(C b (), C b ()). iii. If µ() < and J L (, L ()) then K J L(L p (), L p ()), for all 1 p. Proof. i. From Proposition we have the result. ii. If J C b (, L 1 ()) then, thanks to the previous Proposition 2.1.1, K J belongs to L(L (), C b ()). Moreover, since C b () L (), we have that K J L(C b (), C b ()). iii. From Proposition we have that K J L(L 1 (), L ()). Moreover, since µ() <, L p () L 1 () and L () L p (), then K J : L p () L 1 () L () L p (). 17

51 2.1.2 Regularity of convolution operators There is a large literature on nonlocal diffusion problems, where = R N and the nonlocal term is the convolution with a function J : R N R. Hence, the convolution operator is given by K J (u) = J u. Some examples in the literature with this operator are [1], [7], [18], [22], [52]. Let R N be a measurable set, (it can be = R N, or just a subset R N ). In this section, we study the regularity of the operator K J (u)(x) = J (x y)u(y)dy. (2.14) where J is a function in L p (R N ), for 1 p. Hence the kernel is given by J(x, y) = J (x y), x, y. (2.15) We want to analyze the spaces where the operator K J, (2.14), is defined depending on the integrability of J, as we have done in Proposition for the operator K J. Let us see below the cases that are obtained from Proposition Corollary For 1 p, let R N be a measurable set, if J L p (R N ), then K J L(L p (), L ()). In particular if µ() <, then K J L(L p (), L q ()), for 1 q. Proof. If J L p (R N ), we have that J defined as in (2.15) satisfies that it belongs to L (, L p ()), since sup x J(x, ) L p () = sup J (x ) L p () J L p (R N ) <. x Thus, thanks to Proposition 2.1.1, we have that K J L(L p (), L ()). In particular, if µ() < then K J L(L p (), L q ()), for all 1 q. On the other hand, if µ() =, (like in the case of = R N ), then K J is not necessarily in L(L p (), L q ()), for q. In the proposition below we prove the cases which can not be obtained as a consequence of Proposition Proposition For 1 p, let R N be a measurable set with µ() =, i. if J L r (R N ) and 1 q = 1 p + 1 r 1 then K J L(L p (), L q ()), and In particular, if r = 1 we can take p = q. K J L(L p (),L q ()) J L r (R N ). ii. If R N is open, J W 1,r (R N ) and 1 q = 1 p + 1 r 1 then K J L(L p (), W 1,q ()), and K J L(L p (),W 1.q ()) J W 1,r (R N ). 18

52 Proof. i. We use Young s inequality (see [13, p. 14]) for the convolution, f g L q (R N ) f L r (R N ) g L p (R N ), with 1 q = 1 p + 1 r 1, for 1 p, q. Let us consider the following extension of u, { u(x), if x ũ(x) =, if x /, thus, we have for x K J (u)(x) = Now, we define the extension of the operator K J J (x y)u(y)dy = J (x y)ũ(y)dy = (J ũ) (x). R N as K J (u)(x) = (J ũ) (x), for x R N, then K J (u)(x) = ( KJ (u)) (x), for x. Thanks to Young s inequality, we have K J (u) L q () K J (u) L q (R N ) J L r (R N ) ũ L p (R N ) = J L r (R N ) u L p (). Hence, K J (u) L q () J L r (R N ) u L p (), for all p, q, r such that 1 q = 1 p + 1 r 1. ii. Following the same arguments made in Proposition in (2.1), we know that for x, ( K J (u) = K J (u) = K J (u)) xi x i x i Then, applying part i. to K J (u) L q () and K J x i (u) L q () we have that for p, q, r such that 1 q = 1 p + 1 r 1, K J L(L p (), W 1,q ()). Thus, the result Compactness of K J Under the hypotheses on J in Proposition 2.1.1, in this section we give a result of compactness of the operator K J. The following lemma is a well known characterization of compact operators (see [13, p.157]). Lemma Let E, F be Banach spaces and (T n ) n N be a sequence of operators with finite rank from E to F, and let T L(E, F ) such that T n T L(E,F ), as n goes to. Then T K(E, F ), that is, T is compact. The lemma below will help us to apply the previous Lemma to the operator K J. Lemma For 1 q < and 1 p, let (, µ) be a measure space, then any function H L q (, L p ()) can be approximated by functions of separated variables in L q (, L p ()). 19

53 Proof. Observe that if h(x, y) = f(x)g(y) is a function with separated variables, with f L q () and g L p (), then h L q (, L p ()). We consider the space V = {Finite linear combinations of functions with separated variables } L q (, L p ()). First we prove that V is densely included in L q (, L p ()), for 1 q < and 1 < p (i.e., all the cases except p = 1). To prove this, we suppose that there exists a function ( ) ( ( ) ) L L q (, L p ()) = L q, L p (), with L V, then, M j=1 M j=1 L(x, y) M f j (x)g j (y)dx dy =, M N, f j L q (), g j L p () j=1 L(x, y)f j (x)g j (y)dx dy =, M N, f j L q (), g j L p () g j (y) L(x, y)f j (x)dx dy =, M N, f j L q (), g j L p (). In particular, if we fix f L q (), g(y) L(x, y)f(x)dx dy = then Therefore g L p (), L(x, y)f(x)dx = f L q (), for a.e. y. L(x, y) = for a.e. x, y. With this, we have proved that V L q (, L p ()) densely, for 1 q < and 1 < p. Now we prove that V is densely included in L q (, L p ()), for 1 q < and p = 1. To prove this, we follow the same arguments we have already used, then we suppose that there exists a function L (L q (, L ())) = L q (, M()), with L V, where we denote M() as the set of Radon measures (see Theorem 1.1.4). Then L L q (, M()) is defined as Since L V, we have that M j=1 M j=1 j=1 x L(x, ) M(). M f j (x)g j (y)d y L(x, y) dx =, M N, f j L q (), g j L () f j (x)g j (y)d y L(x, y) dx =, M N, f j L q (), g j L () f j (x) g j (y)d y L(x, y) dx =, M N, f j L q (), g j L (). 2

54 In particular, if we fix g L (), f(x) g(y)d y L(x, y) dx = then Therefore f L q (), g(y)d y L(x, y) = g L (), for a.e. x. L(x, ) = for a.e. x, thus L With this, we have proved that V L q (, L ()) densely, for 1 q < and p = 1. In the following proposition we prove the main result of compactness. Note that we have almost the same assumptions as in Proposition 2.1.1, (see Remark 2.1.8). Proposition i. For 1 p and 1 q <, if J L q (, L p ()) then K J L(L p (), L q ()) is compact. ii. For 1 p, if J BUC(, L p ()) (bounded and uniformly continuous), then K J L(L p (), C b ()) is compact. In particular, K J L(L p (), L ()) is compact. iii. For 1 p and 1 q <, if R N is open and J W 1,q (, L p ()) then K J L(L p (), W 1,q ()) is compact. Proof. i. Since J L q (, L p ()), for 1 p and 1 q <, we know from Lemma that there exist M(n) N and f n j L q (), g n j L p () with j = 1,..., M(n) such that J(x, y) can be approximated by functions that are a finite linear combination of functions with separated variables defined as, J n (x, y) = M(n) j=1 f n j (x)g n j (y) and J J n L q (,L p ()), as n goes to. First of all we are going to prove that K J can be approximated by operators with finite rank in L(L p ), L q ()). To do this, we first define and we prove that K n J K n J (u)(x) = K J n(u)(x) = M(n) j=1 converges strongly to the operator K J (u)(x) = J(x, y)u(y)dy. fj n (x) gj n (y)u(y)dy, Since K J KJ n = K J J n, and thanks to Proposition 2.1.1, we have that, K J K n J L(L p (),L q ()) J J n L q (,L p ()). 21

55 Therefore, thanks to Lemma 2.1.6, we have that J J n L q (,L p ()), as n goes to. Hence, we have proved that K J K n J L(L p (),L q ()), as n goes to. Finally applying Lemma to the operator K J, we have that K J L (L p (), L q ()) is compact. ii. If J BUC(, L p ()), then hypothesis (2.5) of Proposition is satisfied and then K J L(L p (), C b ()). Now, we consider u B L p (), where B is the unit ball in L p (). Now, we prove using Ascoli-Arzela Theorem (see [13, p. 111]), that K J (B) is relatively compact in C b (). Let x, z, u B, thanks to Hölder s inequality, we have, K J (u)(z) K J (u)(x) = J(z, y)u(y)dy J(x, y)u(y)dy J(z, ) J(x, ) L p () u (2.16) L p () J(z, ) J(x, ) L p (). Since J BUC(, L p ()), then for all ε >, there exists δ > such that if x, z satisfy that d(z, x) < δ, then J(z, ) J(x, ) L p () < ε. Hence, we have that K J(B) is equicontinuous. On the other hand, thanks to Hölder s inequality, x, K J (u)(x) = J(x, y)u(y)dy J(x, ) L p () <, u B. Thus, the hypotheses of Ascoli-Arzela Theorem are satisfied, then we have that K J (B) is precompact. Therefore we have proved that K J L(L p (), C b ()) is compact. The second part of the result is immediate. We have proved that K J L(L p (), C b ()) is linear and compact. Moreover, C b () L (), then we have that K J L(L p (), L ()) is compact. iii. Thanks to the argument (2.1) in Proposition 2.1.1, we have that x i K J (u) = K J (u). Since J W 1,q (, L p ()), we have that J L q (, L p ()) and J x x i L q (, L p ()), i for all i = 1,..., N. Using part i. we obtain that K J x i L (L p (), L q ()) is compact. Thus, if B is the unit ball in L p (), we have that K J (B) and K J x i (B) are precompact for all i = 1,..., N. Now we consider the mapping T : L p () ( (L q ()) N+1 ) u K J (u), K J (u),..., K J (u). x 1 x N Thanks to Tíkhonov s Theorem (see [42, p. (L q ()) N ]), we know that T (B) is precompact in 22

56 Moreover, we consider the mapping S : W 1,q () ( (L q ()) N+1 ) g g, g g x 1,..., x N. Since S is an isometry, i.e., g W 1,q () = S( g) (L q ()) N+1, then we have that S 1 Im(S) : Im(S) (L q ()) N+1 W 1,q () is continuous. On the other hand, thanks to the hypotheses on J and Proposition 2.1.1, we have that K J L ( L p (), W 1,q () ). Thus, Im(T ) Im(S). Hence, the operator K J : L p () W 1,q (), can be written as K J (u) = S 1 Im(S) T (u). Therefore, we have that K J is the composition of a continuous operator S 1 Im(S), with a compact operator T. Thus, the result. Remark In general, we have proved that K J is compact, under the same hypotheses of Proposition But to prove that K L(L p (), L ()) is compact, we assume that J BUC(, L p ()), instead of J L (, L p ()), that was the assumption in Proposition Moreover we have not proved that K J L(L p (), W 1, ()) is compact. We finish this section applying interpolation theorems. The following result is valid for a general operator K, not necessarily an integral operator. Proposition Let (, µ) be a measure space, and let µ() <. For 1 p < p 1 <, if K L(L p (), L p ()) and K L(L p 1 (), L p 1 ()) then K L(L p (), L p ()), for all p [p, p 1 ]. Suppose that either: i. K L(L p (), L p ()) is compact, ii. K L(L p 1 (), L p 1 ()) is compact, then K L(L p (), L p ()) is compact for all p [p, p 1 ]. Proof. Thanks to the hypotheses and Riesz-Thorin Theorem, (see [1, p. 196]), we have that K L(L p (), L p ()), for all p [p, p 1 ]. The proof of the compactness can be seen in [21, p. 4] Positiveness of the operator K J In this section, given a nonnegative function z, which is not identically zero, we describe the set of points where K J (z) is strictly positive, under hypothesis (2.3) on the kernel J. To do this, we need first to introduce the definition of essential support associated to a nonnegative measurable function z. 23

57 Definition Let z be a measurable nonnegative function z : R. We define the essential support associated to z as: P (z) = { x : δ >, µ ( {y : z(y) > } B δ (x) ) > }, (2.17) where µ is the measure of the set, and B δ (x) is the ball centered in x, with radius δ. The following lemma will be useful to understand better the essential support of a nonnegative function z. Lemma Let z be a nonnegative measurable function z : R, then the following properties are equivalent: i. z not identically zero. ii. P (z). iii. µ (P (z)) >. Proof. i) ii) From Definition of P (z), we have that x / P (z), if and only if, there exists δ > such that µ ( {y : z(y) > } B δ (x) ) =. (2.18) Then, we have that B δ (x) P (z) c. Indeed, since B δ (x) is open then for any x B δ (x), there exists ε >, such that B ε ( x) B δ (x). Thus, from (2.18) we obtain that µ ( {y : z(y) > } B ε ( x) ) =. Hence x P (z), and we have proved that B δ (x) P (z) c. This implies that, P (z) c is open and z(x) = for a.e. x P (z) c. Furthermore, we have that P (z) c is the largest open set where z almost everywhere. Now, we assume that P (z) =, then P (z) c =. Thus z a.e. in, and we arrive to contradiction. Therefore, P (z). i) iii) If µ(p (z)) =, then P (z) c = \ P (z) satisfies that µ(p (z) c ) = µ(). Hence, z a.e. in, and we arrive to contradiction. Therefore, µ(p (z)) >. ii) i) If P (z) then, there exists x and δ > such that µ ({y : z(y) > } B δ (x)) >. Thus, there exists a set with positive measure where z is strictly positive. iii) ii) If µ(p (z)) >, then P (z). Let us introduce the following definitions. 24

58 Definition Let z be a measurable nonnegative function z : R. For R >, we denote P (z) = P (z), the essential support of z, and we define the open sets: PR(z) 1 = B(x, R), PR(z) 2 = B(x, R),..., PR(z) n = B(x, R),... for all n N. x P (z) x P 1 R (z) x P n 1 R (z) Remark If the metric of is equivalent to the euclidean metric in R N, then the sets PR n (z) in Definition are equal to P n R(z) = (P (z) + B nr ), where B nr is the ball centered in zero with radius nr. Definition Let (, µ, d) be a metric measure space, and R >. We say that is R-connected if x, y, N N and a finite set of points {x,..., x N } in such that x = x, x N = y and d(x i 1, x i ) < R, for all i = 1,..., N. Lemma If is compact and connected then is R-connected for any R >. Proof. Since is compact, given R >, there exists n N and {y 1,..., y n } such that n B(y i, R/4). Moreover, i = 1,..., n, B(y i, R/4) B(y j, R/4), since otherwise i=1 j i = B(y i, R/4) B(y j, R/4), contradicting that is connected. Analogously, we have j i that i 1,..., i k {1,..., n}, k r=1 B(y ir, R/4) j {i 1,...,i k } B(y j, R/4) Now, let us prove that is R-connected for any R >. Then, given any two points x, y in, first of all we consider x = x, and we choose a ball such that x B(y i1, R/4), since is connected, then there exists a ball B(y i2, R/4) that intersects B(y i1, R/4), and we choose x 1 = y i2. If y B(y i2, R/4), we finish the proof, if not, following this constructing argument, we obtain there exists a ball B(y i3, R/4) that intersects B(y i1, R/4) B(y i2, R/4), and we choose x 2 = y i3. If y B(y i3, R/4) we finish the proof, if not, with a continuation argument, we find a finite set of points {x,..., x n 1 } such that x = x, x N 1 = y and d(x i 1, x i ) < R, for i = 1,..., N, where N n. Thus, the result. Lemma Let (, µ, d) be a metric measure space such that is R-connected. For a fixed x, and for some R >, we set Px = {x }, PR,x 1 = B(x, R) and PR,x n = B(x, R) for all n N. x P n 1 R,x Then, for every compact set in K, there exists n(x ) N such that K PR,x n for all n n(x ). Furthermore, if is compact, there exists n N such that for any y, = PR,y n, for all n n. 25

59 Proof. Since is R-connected, fixed x, for any y, M = M y N and a finite set of points {x,..., x M } such that x M = y and d(x i 1, x i ) < R, for all i = 1,..., M. Thus, x 1 B(x, R) = PR,x 1, x 2 B(x 1, R) PR,x 2, B(x i, R) P i+1 R,x, for all i = 1,..., M. In particular, y PR,x M and B(y, R) P M+1 R,x. (2.19) Arguing analogously, we obtain that x PR,y M. Then we have proved that if is R-connected, x, y, N N such that x PR,y N and y P R,x N, i.e., there exists and R-chain of N-steps that joins x and y, and there exists and R-chain of N-steps that joins y and x. On the other hand, since K is compact, K y K B(y, R), there exists n N such that K n i=1 B(y i, R). From (2.19), for every y i there exists M yi such that B(y i, R) P My i +1 R,x. We choose n(x ) = max (M y i + 1), and we obtain that K P n(x ) i=1,...,n R,x. Therefore, K = PR,x n, for all n n(x ). Thus, the result. If is compact. From the previous result we know that fixed x, N = N(x ) such that = PR,x N. Moreover, = PR,x N if and only if y, y PR,x N and x PR,y N, i.e., there exists an R-chain of N-steps that joins x and y. Therefore, for all y 1, y 2 there exists an R-chain of 2N-steps that joins y 1 and y 2. This is because there exists an R-chain of N- steps that joins y 1 with x, and there exists an R-chain of N-steps that joins x with y 2, then joining both R-chains, we obtain that for any y 1, y 1 PR,y 2N 2, for all y 2. Hence PR,y 2N 2, for all y 2. Thus, we have proved the result with n = 2N. Now, we prove the main result. Proposition Let (, µ, d) be a metric measure space, and let J satisfy that J not identically zero, with J(x, y)> for all x, y, such that d(x, y)<r, (2.2) for some R >. If z is a measurable function defined in, with z, not identically zero. Then, P (K n J (z)) P n R(z), for all n N. If is R-connected, then for any compact set K, n (z) N, such that P (K n J (z)) K, for all n n (z). If is compact and connected, then n N, such that, for all z measurable and not identically zero P (K n J (z)) =, for all n n. Proof. First of all we prove that P (K J (z)) PR 1 (z). Since z, not identically zero, and as a consequence of Lemma , we have that µ (P (z)) >. Then, K J (z)(x) = J(x, y)z(y)dy 26 P (z) J(x, y)z(y)dy.

60 From hypothesis (2.2) on the positivity of J, we have that K J (z)(x) > for all x B(y, R) = PR(z). 1 (2.21) y P (z) Since PR 1(z), is an open set in, we have that, if x P R 1 (z), then Thus, thanks to (2.21) and (2.22), we have that µ ( B(x, δ) P 1 R(z) ) > for all < δ R. (2.22) P (K J (z)) P 1 R(z). (2.23) Applying K J to K J (z), following the previous arguments and thanks to (2.23), we obtain P ( K 2 J(z) ) P 1 R(K J (z))= x P (K J (z)) Therefore, iterating this process, we finally obtain that B(x, R) B(x, R) = PR(z). 2 x P 1 R (z) P (K n J (z)) P n R(z), n N. (2.24) Now consider K a compact set in, and taking x P (z), then thanks to Lemma there exists n (z) N, such that K P n R (z) for all n n, then thanks to (2.24), K P (K n J (z)) for all n n. If is compact and connected, thanks to Lemma , is R- connected. From Lemma there exists n N such that for any y, = PR,y n, for all n n. Hence, from (2.24), for any z not identically zero, taking y P (z), P (KJ n(z)) P R,y n =, for all n n. Remark In Figure 2.1 can be seen which is the set where the function J is strictly positive under hypothesis (2.2), in the particular case in which R. Figure 2.1: Domain where J is strictly positive if R. 27

61 Furthermore, the hypothesis (2.2) is somehow an optimal condition. We give below a counterexample in R: if the hypothesis (2.2) is not satisfied, we find a function z, for which the previous Proposition is not satisfied. Counterexample: Let R, = [, L], with L > and let us fix an arbitrary x = 1/2 [, 1], and R > small enough such that (1/2 R, 1/2 + R) [, 1]. We consider a function J satisfying that J defined as { 1, (x, y) ( [, 1] [, 1] ) \ (( 1 J(x, y) = 2 R, R) ( 1 2 R, R)), with d(x, y)<r, for the rest of (x, y). (2.25) We remark that, J(x, y) =, (x, y) ( 1 2 R, R) ( 1 2 R, R). Now, we consider a function z : R, z, such that P (z ) [1/2, 1]. Since z (y) = for all y [1/2, 1], we have that K J (z )(x) = J(x, y)z (y)dy = J(x, y)z (y)dy = P (z ) 1 1/2 J(x, y)z (y)dy. Moreover, from (2.25), we have that for x [, 1/2), J( x, y) = for all y [1/2, 1]. Let us prove this below: Thus, Hence If x (1/2 R, 1/2), then if y [1/2, 1/2+R), then ( x, y) ( 1 2 R, 1 2 +R) ( 1 2 R, 1 2 +R), and J( x, y) = ; if y [1/2 + R, 1], then d( x, y) > R, and J( x, y) =. If x [, 1/2 R], then for y [1/2, 1], d( x, y) > R, and J( x, y) =. K J (z )( x) = 1 1/2 If we apply K J to K J (z ), we obtain that KJ(z 2 )(x) = J(x, y)k J (z )(y)dy = J( x, y)z (y)dy =, x [, 1/2). P (K J (z )) [1/2, 1]. P (K J (z )) J(x, y)k J (z )(y)dy = 1 1/2 J(x, y)k J (z )(y)dy. Arguing as above, we have that for any x [, 1/2), J( x, y) =, for all y [1/2, 1]. Thus, P ( K 2 J(z ) ) [1/2, 1]. Therefore, iterating this process, we obtain that P (K n J (z )) [1/2, 1] for all n N. Hence P (K n J (z )) [, 1] for all n N, and the hypothesis (2.2) is essentially optimal. 28

62 2.1.5 The adjoint operator of K J In this section we describe the adjoint operator associated to K J, and we prove that if J L 2 ( ) and J(x, y) = J(y, x) then the operator K J is selfadjoint in L 2 (). Proposition For 1 p <, 1 q <. Let (, µ) be a measure space. We assume that the mapping x J(x, ) satisfies that J L q (, L p ()), and the mapping y J(, y) satisfies that J L p (, L q ()). Then the adjoint operator associated to K J L(L p (), L q ()), is K J : L q () L p (), with K J = K J, where J (x, y) = J(y, x). If J satisfies that J(x, y) = J(y, x), (2.26) then for u L p () and v L q (), K J (u), v L q,l = u, K J(v) q L p,lp. (2.27) In the particular case in which p = q = 2 and J L 2 ( ), the operator K J is selfadjoint in L 2 (). Proof. We consider u L p () and v L q (). Thanks to Fubini s Theorem and the hypothesis on J and K J (u), v L q (),L q () = J(x, y)u(y)dy v(x)dx= J(x, y)v(x)dx u(y)dy, J(x, y)v(x)dx u(y)dy = u, K J(v) L p (),L p (), with KJ(v)(y) = and J (y, x) = J(x, y). J(x, y)v(x)dx = J (y, x)v(x)dx = K J (v)(y), In particular if u L p () and v L q () and J satisfies that J(x, y) = J(y, x), we obtain K J (u), v L q,l = u, K J(v) q L p,lp. (2.28) An immediate consequence of (2.28) is the case in which p = q = 2, that we have that K J is selfadjoint in L 2 (). 29

63 2.1.6 Spectrum of K J In this section, we are going to prove that under certain hypotheses on K J, σ X (K J ) is independent of X, with X = L p (), where 1 p or X = C b (). We also characterize the spectrum of K J when K J is selfadjoint in L 2 (), and we finish this section proving that under the same hypothesis on the positivity of J in Proposition , the spectral radius of K J in C b () is a simple eigenvalue that has a strictly positive eigenfunction associated. The proposition below is for a general compact operator K, not only for the integral operator K J (see Propositions to check compactness for operators with kernel, K J ). Proposition Let (, µ, d) be a metric measure space with µ as in Definition and µ() <. i. For 1 p < p 1 <, if K L(L p (), L p 1 ()) and additionally K L(L p (), L p ()) is compact then K L(L p (), L p ()), p [p, p 1 ], and σ L p(k) is independent of p. ii. For 1 p < p 1, if K L(L p (), L p 1 ()) is compact, then K L(L p (), L p ()), p [p, p 1 ], and σ L p(k) is independent of p. iii. For 1 p, if K L(L p (), C b ()) is compact and X = C b () or X = L r () with r [p, ], then K L(X, X), and σ X (K) is independent of X. Proof. i. Thanks to Proposition 2.1.9, we have that K L(L p (), L p ()) is compact for all p [p, p 1 ]. Thus the spectrum of K is composed by zero and a discrete set of eigenvalues of finite multiplicity, (see [13, chap. 6]). Let us prove now that the eigenvalues of the spectrum σ L p ()(K) are independent of p. We prove first that σ L p 1 () σ L p (): if λ σ L p 1 (K) is an eigenvalue, then the associated eigenfunction Φ L p 1 (). Since µ() < we have that L p 1 () L p () continuously, for all p p, then Φ L p (). Thus we obtain that λ σ L p(k) for all p [p, p 1 ]. Now, we prove that σ L p () σ L p 1 () : if λ σ L p ()(K) is an eigenvalue, with p [p, p 1 ), then the associated eigenfunction Φ L p () satisfies that K(Φ) = λφ. (2.29) Since L p () L p () continuously and K : L p () L p 1 (), then K(Φ) L p 1 (). From (2.29), we obtain that Φ L p 1 (). Hence, Φ L p () for p [p, p 1 ]. Thus, the result. ii. We know that K L(L p (), L p 1 ()) is compact, and we have that K : L p 1 () L p () L p 1 () and K : L p () L p 1 () L p (). Therefore K L(L p 1 (), L p 1 ()) is compact, and the hypotheses of Proposition are satisfied. Therefore K L(L p (), L p ()) is compact for all p [p, p 1 ]. From part i., we 3

64 have the result. and iii. We know that K L(L p (), C b ()) is compact. Since µ() <, we have that and for r [p, ] K : C b () L p () C b () K : L p () C b () L p () K : L r () L p () C b () L r () Therefore, K L(X, X) is compact for X = C b () or X = L r () with r [p, ]. Hence, following the arguments in i. we have that σ X (K) is independent of X. The following result holds for a general selfadjoint operator in a Hilbert space, and the proof can be found in [13, p. 165]. Proposition Let H be a Hilbert space and T L(H) a selfadjoint operator. Take m = inf u H u H =1 T u, u H and M = sup T u, u H. u H u H =1 Then σ(t ) [m, M] R, m σ(t ) and M σ(t ). We can apply this Proposition to the operator K J, obtaining more details about its spectrum. Proposition Let (, µ, d) be a metric measure space with µ() <. We assume K J L(L p (), C b ()) is compact, and p 2. Let X = L p (), with p [p, ], or X = C b (), and J satisfies that J(x, y) = J(y, x). Then K J L(X, X) and σ X (K J ) \ {} is a real sequence of eigenvalues of finite multiplicity, independent of X, that converges to. Moreover, if we consider m = inf K J (u), u L u L 2 2 () and M = sup K J (u), u L 2 (), () u L 2 () u L 2 =1 () u L 2 =1 () (2.3) then σ X (K J ) [m, M] R, m σ X (K J ) and M σ X (K J ). In particular, L 2 () admits an orthonormal basis consisting of eigenfunctions of K J. Proof. Thanks to Proposition , K J is selfadjoint in L 2 (), then σ L 2(K J ) \ {} is a real sequence of eigenvalues of finite multiplicity that converges to, (see [13, chap.6 ]). Furthermore, from Proposition we have that σ X (K J ) is independent of X. Thus, the result. On the other hand, as a consequence of Proposition , we have that σ X (K J ) [m, M] R, with m σ X (K J ) and M σ X (K J ), where m and M are given by (2.3). Thanks to the Spectral Theorem (see [13, chap.6]), we know that L 2 () admits an orthonormal basis consisting of eigenfunctions of K J. 31

65 The following Corollary states that under the hypotheses of Proposition , any nonnegative eigenfunction associated to the operator K J is in fact strictly positive positive as well as its associated eigenvalue. Corollary Let (, µ, d) be a metric measure space, let J satisfy the hypotheses of Proposition and assume is R-connected. If Φ, not identically zero, is an eigenfunction associated to an eigenvalue λ of the operator K J, then Φ >, and the eigenvalue, λ, is also strictly positive. Proof. Thanks to Proposition , we know that, for every function Φ, not identically zero defined in, it happens that P (KJ n(φ)) P R n (Φ), n N. On the other hand, since Φ is an eigenfunction associated to an eigenvalue λ of the operator K J, we have that KJ n(φ) = λn Φ, n N. Moreover, from Proposition , we know that for any compact set K, there exists n N such that P (KJ n(φ)) K for all n n. Thus, KJ n(φ) = λn Φ is strictly positive in K for all n n. Therefore Φ must be strictly positive in any compact set K of. Hence, λ > and Φ must be strictly positive in. Now, let us give some results about the spectral radius of the operator K, where the spectral radius is r(k) = sup σ(k). To give these properties about the spectral radius we will use Kreĭn- Rutman Theorem. The definitions below will be helpful to introduce the following results, (see [51], [38]). Definition i. A real Banach Space X is called ordered if there exists a given closed convex cone C in X (with the vertex at the origin) satisfying C ( C) = {}, i.e. C X is closed, and α, β [, ) and x, y C = αx + βy C; x C, x C = x = C. Then C is called the positive cone of X. This is equivalent to say that x C if and only if x ; and x y if and only if x y. ii. If C has no empty interior, Int(C), in X, then X is called strongly ordered. iii. In a strongly ordered space, an everywhere defined linear operator T : X X is called strongly positive if there exists n N such that T n (C \ {}) Int(C), for all n n. Theorem (Kreĭn-Rutman Theorem) Let X be a strongly ordered Banach space with positive cone C. Assume that T : X X is a strongly positive compact linear operator on X. Then i. the spectral radius of T, r(t ) = sup σ(t ), is a positive, simple eigenvalue of T ; ii. the eigenfunction u in X\{} associated with the eigenvalue r(t ) can be taken in Int(C); 32

66 iii. if µ is in the spectrum of µ < r(t ); T, µ r(t ), then µ is an eigenvalue of T satisfying iv. if µ is an eigenvalue of T associated with an eigenfunction v in C \ {} then µ = r(t ). To apply the Kreĭn-Rutman Theorem to the operator K J, we work in the space C b (), with compact, and we consider the positive cone C = {f ; f C b ()}, with Int(C) = {f C b (); f(x) >, x }. Thus, in the proposition below, we prove that the spectral radius of the operator K is a simple eigenfunction that has an associated eigenfunction that is strictly positive. Proposition Let (, µ, d) be a metric measure space, with compact and connected. We assume that J satisfies J(x, y) = J(y, x) and J(x, y) >, x, y such that d(x, y) < R, for some R >, and K J L(L p (), C b ()), with 1 p, is compact, (see Proposition ii.). Then K J L(C b (), C b ()) is compact, the spectral radius r Cb ()(K J ) is a positive simple eigenvalue, and its associated eigenfunction is strictly positive. Proof. Since is compact and connected then from Proposition we obtain that, there exists n N such that, for any nonnegative u C b (), = P n R (u), for all n n, (see Definition ), and we know that for every nonnegative u C b () and n N, P (K n (u)) P n R (u). Therefore = P n R (u) P (Kn (u)) for all n n, i.e., for any nonnegative u C b (), K n J (u) > in for all n n. Hence, K J is strongly positive in C b (). Moreover K J : C b () L p () C b () is compact. Thus, we have that all hypotheses of Kreĭn-Rutman Theorem are satisfied in the space C b () for the operator K J, then the spectral radius r Cb ()(K J ) is a positive simple eigenvalue with an eigenfunction Φ associated to it that is strictly positive. 2.2 The multiplication operator hi Let h be a function defined in, h : R. In this section be will focus in the study of the linear multiplication operator hi, that maps u(x) h(x)u(x). We will start studying the spaces where the operator is defined depending on the integrability or continuity of the function h. In particular, we are interested in the multiplication operator hi with h L () or h C b (). We will describe which is its adjoint operator, and we will also prove that if h L (), then the operator hi is selfadjoint in L 2 (), and we finish describing the spectrum and resolvent set of hi. The following proposition studies the regularity of hi. 33

67 Proposition Let (, µ, d) be a metric measure space. i. For 1 p, q, if h L r (), and if 1 p + 1 r = 1 q, then hi L(Lp (), L q ()), and hi L(L p (),L q ()) h L r (). ii. If h L () then hi L (L p (), L p ()), for all 1 p, and hi L(L p (),L p ()) h L (). iii. If h C b (), let X = L p (), with 1 p or X = C b (), then hi L (X, X), and hi L(X,X) h Cb (). Proof. i. Thanks to Hölder s inequality, and the fact that h L r () and u L p () h u q L q () = h(x)u(x) q dx ( ) 1/α ( h(x) qα dx u(y) qβ dy = h q L r () u q L p (), ) 1/β with 1 α + 1 β = 1, qα = r and qβ = p, then p, r and q have to satisfy that 1 p + 1 r = 1 q. ii. For 1 p <, we consider u L p (). Since h L () we have that h u p L p () = h(x)u(x) p dx h p L () u p L p (). For p =, we consider u L (). Since h L () we have that Thus, the result. h u L () = sup h(x)u(x) h L () u L (). x iii. Since C b () L (), then from ii., we have the result for X = L p (). Now, if u C b (), then hu is continuous. Furthermore, we have that Thus, the result. h u Cb () = sup h(x)u(x) h Cb () u Cb (). x Lemma Let (, µ, d) be a metric measure space, then i. hi L(L p (), L p ()), for 1 p if and only if h L (). ii. hi L(C b (), C b ()) if and only if h C b (). 34

68 Proof. i. Thanks to Proposition 2.2.1, we know that if h L () then hi belongs to L(L p (), L p ()). Let us see the converse implication. Since hi L(L p (), L p ()), there exists < C R such that hu L p () C u L p (), u L p (). (2.31) Now, we argue by contradiction. Assume h / L (), then for all k R, the exists a set A k, such that µ(a k ) >, where h(x) > k, x A k. Then for any < k R we can choose u k L p () such that u k L p () = u k L p (A k ). From (2.31), we have ( 1/p C u k L p () h u k dx) p > k u k L p (A k ). Thus C > k, for any k >. Hence, we arrive to contradiction, and h L (). ii. Thanks to Proposition 2.2.1, we know that if h C b () then hi L(C b (), C b ()). Let us see the converse implication. The boundedness is obtained from part i.. Moreover, since hi L(C b (), C b ()), if we choose u 1, then hu = h C b (). Thus, the result. Remark Let X = L p (), with 1 p or X = C b (). In general, if h L () is not identically zero, then the operator h I : X X is not compact. For instance, in the particular case in which the function h(x) = 1, x, we have that hi is the identity operator, and the identity is not compact. This is because the unit ball in X is not compact, since the dimension of X is infinity. The following result describes the adjoint operator of the multiplication operator hi, and we prove that it is selfadjoint in L 2 (). Proposition Let (, µ) be a measure space and let h L () then the adjoint operator associated to hi L(L p (), L p )), with 1 p <, is (hi) : L p () L p (). where (hi) = hi. In particular if p = 2, hi is selfadjoint in L 2 (). Proof. For 1 p <, we have that for h L (), u L p (), and v L p () then, hi(u), v L p, L = h(x)u(x)v(x)dx = u(x)h(x)v(x)dx, p and u(x)h(x)v(x)dx = u, (hi) (v) L p, L p for (hi) (v)(x) = h(x)v(x). Thus (hi) = hi, but in this case hi : L p () L p (). It is immediate that if p = 2, then hi is selfadjoint. 35

69 Now, we give a description of the spectrum and the resolvent set of the multiplication operator hi L(X, X), with X = L p (), for 1 p or X = C b (). We denote as EV (hi) the eigenvalues of the multiplication operator hi, and Im(h) R the range of h. Proposition Let (, µ, d) be a metric measure space. i. If X = L p (), with 1 p, we assume h L. ii. If X = C b (), we assume h C b (). The resolvent set of the multiplication operator is given by ρ X (hi) = C \ Im(h), and its spectrum is σ X (hi) = Im(h), and they are independent of X. Moreover, for X = L p (), the eigenvalues associated to hi exist only when the function h is constant in subsets of with positive measure, i.e., EV (hi) = {α ; µ ({x ; h(x) = α}) > ). The eigenvalues of the multiplication operator hi have infinite multiplicity. For X = C b (), EV (hi) {α ; A open with µ(a) > such that A {x ; h(x) = α}} = F and the eigenvalues of hi in F have infinite multiplicity. Proof. i. Thanks to Lemma 2.2.2, we know that h L () if and only if hi L(L p (), L p ()), for all 1 p. We consider f L p () and u L p (), then h(x)u(x) λu(x) = f(x) (h(x) λ)u(x) = f(x) u(x) = f(x) h(x) λ = 1 h(x) λ f(x). (2.32) Then we have that λ ρ L p ()(hi) if and only if (hi λi) 1 L(L p (), L p ()), and thanks to Lemma 2.2.2, (hi λi) 1 L(L p (), L p 1 ()) if and only if h λ L (), and this happens when 1 h(x) λ C, x, then, there exists δ >, such that h(x) λ > δ, x, i.e., if and only if λ Im(h). Then, we have proved that ρ L p ()(hi) = C \ Im(h) and its spectrum is by definition σ(hi) = C \ ρ(hi) = Im(h). Since Im(h) is independent of L p (), then the spectrum of hi L(L p (), L p ()) is independent of p. 36

70 The eigenvalues of hi satisfy by definition that there exists Φ L p () with Φ, such that h(x)φ(x) = λφ(x) and this only happens if there exists a set A, with µ(a) >, such that h(x) = λ for all x A. Then, the eigenfunctions Φ associated to λ satisfy that Φ L p (A), and Φ(x) =, x \ A. Hence, we have that Ker(hI λi) = L p (A). Thus, the result. ii. Thanks to Proposition 2.2.1, since h C b () we have that hi L(X, X). The rest of the proof follows the same arguments as in i.. Moreover, if there exists an open set A, with µ(a) >, such that h(x) = λ for all x A, then λ is an eigenvalue of hi in C b (), and the space of eigenfunctions associated to λ is given by Ker(hI λi) = {Φ C b () : Φ(x) =, x \ A}, which has infinite dimension. Thus, the result. 2.3 Green s formulas for K J h I In this section we introduce the Green s formulas for K J h I, where h (x) = J(x, y)dy. We will assume that h L (), and this is satisfied if and only if J L (, L 1 ()). Green s formulas will be useful to obtain some properties of the sign of the spectrum of the operator K J hi. Proposition (Green s formulas) Let (, µ, d) be a metric measure space such that µ() <. If J L p (, L p ()), for 1 p <, and h L (), and if then for u L p () and v L p (), K J (u) h I(u), v L p,l p = 1 2 In particular, if p = 2 we have that for u L 2 () K J (u) h I(u), u L 2,L 2 = 1 2 J(x, y) = J(y, x), (2.33) J(x, y)(u(y) u(x))(v(y) v(x))dy dx. (2.34) J(x, y)(u(y) u(x)) 2 dy dx. (2.35) Proof. We denote the integral term of the right hand side of (2.34) by I 1 = J(x, y)(u(y) u(x))(v(y) v(x))dy dx = J(x, y)(u(y) u(x))v(y)dy dx J(x, y)(u(y) u(x))v(x)dy dx. 37

71 Relabeling variables in the first term of the sum, we obtain I 1 = J(y, x)(u(x) u(y))v(x)dx dy J(x, y)(u(y) u(x))v(x)dy dx. Now, since J(x, y) = J(y, x), I 1 = J(x, y)(u(x) u(y))v(x)dx dy J(x, y)(u(y) u(x))v(x)dy dx. Thanks to Fubini s Theorem, we have that = 2 J(x, y)(u(y) u(x))v(x)dy dx. I 1 Therefore, we have proved that the integral term of the right hand side of (2.34) is equal to J(x, y)(u(y) u(x))(v(y) v(x))dy dx = 2 J(x, y)(u(y) u(x))v(x)dy dx. (2.36) On the other hand, thanks to the hypothesis on J, h L () and from Propositions and 2.2.1, we have that K J h I L(L p (), L p ()), for all 1 p. Hence, if u L p () and v L p () K J (u) h I(u), v L p,l p = = ( ) J(x, y)u(y)dy J(x, y)dy u(x) v(x) dx J(x, y)(u(y) u(x))v(x)dy dx. (2.37) Hence, from (2.36) and (2.37), we obtain (2.34). The second part of the proposition is an immediate consequence of (2.34). 2.4 Spectrum of the operator K hi Let (, µ, d) be a metric measure space. In this section we describe the spectrum of K hi L(X, X), and we prove that, under certain conditions on the operator K, it is independent of X. Moreover, we give conditions on J and h under which the spectrum of K J hi is nonpositive. We start introducing some definitions used in the following theorems, that will be useful to give a description of the spectrum of K hi. Definition If T is a linear operator in a Banach space Y, a normal point of T is any complex number which is in the resolvent set, or is an isolated eigenvalue of T of finite multiplicity. Any other complex number is in the essential spectrum of T. To describe the spectrum of K hi, we use the following theorem that can be found in [34, p. 136]. 38

72 Theorem Suppose Y is a Banach space, T : D(T ) Y Y is a closed linear operator, S : D(S) Y Y is linear with D(S) D(T ) and S(λ T ) 1 is compact for some λ ρ(t ). Let U be an open connected set in C consisting entirely of normal points of T, which are points of the resolvent of T, or isolated eigenvalues of T of finite multiplicity. Then either U consists entirely of normal points of T + S, or entirely of eigenvalues of T + S. Remark If S : Y Y is compact, Theorem implies that the perturbation S can not change the essential spectrum of T. The next theorem describes the spectrum of the operator K hi in X. Theorem Let (, µ, d) be a metric measure space. If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If K L (X, X) is compact, (see Proposition 2.1.7), then σ(k hi) = Im( h) {µ n } M n=1, with M N { }. If M =, then {µ n } n=1 accumulates in Im( h). is a sequence of eigenvalues of K hi with finite multiplicity, that Proof. With the notations of Theorem 2.4.2, we consider the operators S = K and T = hi. First of all, we prove that C \ Im( h) ρ(k hi). We choose the set U in Theorem as U = ρ( hi) = ρ(t ) = C \ Im( h) that is an open, connected set. Since U = ρ(t ), every λ U is a normal point of T. On the other hand, if λ ρ(t ), then (T λ ) 1 L(X, X), and S = K is compact. Then, we have that S(λ T ) 1 L(X, X) is compact. Thus, all the hypotheses of Theorem are satisfied. Now, thanks to Theorem 2.4.2, we have that U = C \ Im( h) consists entirely of eigenvalues of T + S = K hi or U consists entirely of normal points of T + S = K hi. If U = C \ Im( h) consists entirely of eigenvalues of T + S = K hi, we arrive to contradiction, because the spectrum of K hi is bounded. So U = C \ Im( h) has to consist entirely of normal points of T + S. Then, they are points of the resolvent or isolated eigenvalues of T + S = K hi. Since any set of isolated points in C is a finite set, or a numerable set, we have that the isolated eigenvalues are {µ n } M n=1, with M N or M =. Moreover, since the spectrum of K hi is bounded, if M = then {µ n } n=1 of eigenvalues of K hi with finite multiplicity, that accumulates in Im( h). is a sequence 39

73 Now we prove that Im( h) σ(k hi). We argue by contradiction. Suppose that there exists a λ Im( h) that belongs to ρ(k hi). Since the resolvent set is open, there exists a ball B ε ( λ) centered in λ, that is into the resolvent of K hi. Then U = B ε ( λ) is an open connected set that consists of normal points of K hi. With the notation of Theorem 2.4.2, we consider the operators T = K hi and S = K and the open, connected set U = B ε ( λ). Arguing like in the previous case, if λ ρ(t ), we have that S(λ T ) 1 is compact, thus the hypotheses of Theorem are satisfied. Hence U = B ε ( λ) consists entirely of eigenvalues of T + S = hi or U = B ε ( λ) consists entirely of normal points of T + S = hi. If U = B ε ( λ) consists entirely of eigenvalues of T + S = hi, we would arrive to contradiction, because the eigenvalues of hi are only inside Im( h), and the ball B ε ( λ) is not inside Im( h). So U = B ε ( λ) has to consist of normal points of T + S = hi, so they are points of the resolvent of hi or isolated eigenvalues of finite multiplicity of hi. Since ρ( hi) = C \ Im( h), and λ Im( h), we have that λ has to be an isolated eigenvalue of hi, with finite multiplicity. But from Proposition 2.2.5, we know that the eigenvalues of hi have infinity multiplicity. Thus, we arrive to contradiction. Hence, we have proved that Im( h) σ(k hi). With this, we have finished the proof of the theorem. In the following proposition we prove that the spectrum of K hi is independent of X = L p () with 1 p, or X = C b (). Proposition Let (, µ, d) be a metric measure space with µ() <. i. For 1 p < p 1 <, if K L(L p (), L p 1 ()) and additionally K L(L p (), L p ()) is compact and h L (), then K hi L(L p (), L p ()), p [p, p 1 ], and σ L p(k hi) is independent of p. ii. For 1 p < p 1, if K L(L p (), L p 1 ()) is compact and h L (), then K hi L(L p (), L p ()), p [p, p 1 ], and σ L p(k hi) is independent of p. iii. For a fixed 1 p, if K L(L p (), C b ()) is compact and X = C b () or X = L r () with r [p, ], and h C b (), then K hi L(X, X) and σ X (K hi) is independent of X. Proof. Following the same arguments in Proposition 2.1.2, we have that in any of the cases i., ii., or iii., K L(X, X) is compact, where X = L p () with p p p 1 for the cases i. and ii., and X = L p () with p p, or X = C b () for the case iii.. Then, from Theorem we have that σ X (K hi) = Im( h) {µ n } M n=1, with M N with orm =, where {µ n } n are eigenvalues of K hi, with finite multiplicity n {1,..., M}. 4

74 Since Im( h) is independent of X, we just have to prove that the eigenvalues λ σ X (K hi) satisfying that λ / Im( h) are independent of X. Let λ σ X (K hi) be an eigenvalue such that λ / Im( h). We denote by Φ an eigenfunction associated to λ σ X (K hi), then Since λ / Im( h), then from (2.38) we obtain and K(Φ)(x) h(x)φ(x) = λφ(x) (2.38) Φ(x) = 1 h( ) + λ L (). Thanks to the hypotheses on K, we have 1 K(Φ)(x) (2.39) h(x) + λ 1 h( ) + λ K L(Lp (), L p 1 ()). (2.4) We prove first that σ L p 1 () σ L p (): if λ σ L p 1 (K) is an eigenvalue, then the associated eigenfunction Φ L p 1 (). Since µ() < we have that L p 1 () L p () continuously, for all p p, then Φ L p (). Thus we obtain that λ σ L p(k hi) for all p [p, p 1 ]. Now, we prove that σ L p () σ L p 1 () : if λ σ L p ()(K) is an eigenvalue, with p [p, p 1 ), then the associated eigenfunction Φ L p () satisfies (2.39). Since L p () L p () continuously, then from (2.4), we have that 1 h( )+λ K(Φ) Lp 1 (). Hence, from (2.39), we obtain that Φ L p 1 (). Therefore, Φ L p () for p [p, p 1 ], and we have proved the independence of the spectrum respect the space for the cases i. and ii.. The case iii. is analogous to the previous result, using that h C b () and λ Im( h), then 1 h( ) + λ K(Φ) L(Lp (), C b ()). (2.41) Thus, the result. The following results state hypotheses to know in which cases the spectrum of K J hi is nonpositive. Corollary Let (, µ, d) be a metric measure space with µ() <. For 1 p 1, let X = L p (), with p [1, p 1 ] or X = C b (). We assume that K and h satisfy the hypotheses in Proposition with p 2 and we assume that J is such that J(x, y) = J(y, x). i. If h c, with c R such that c > r(k J ), where r(k J ) is the spectral radius of K J then σ X (K J hi) is real and nonpositive. ii. If h = h = J(x, y)dy L () and h satisfies that h (x) > α > for all x, then σ X (K J hi) is nonpositive and is an isolated eigenvalue with finite multiplicity. Moreover if J satisfies that then {} is a simple eigenvalue. J(x, y) >, x, y such that d(x, y) < R 41

75 iii. If h L () satisfies that h h in, then σ X (K J hi) is nonpositive. Proof. Under the hypotheses and thanks to the previous Proposition 2.4.5, we have that σ X (K hi) is independent of X. Hence the rest of the results will be proved in L 2 (). i. From Proposition and Proposition 2.2.4, we have that K J and hi are selfadjoint operators in L 2 (), then we have that K J hi is a selfadjoint operator in L 2 (). By using Proposition , we know that σ L 2 ()(K J ) is composed by real values that are less or equal to r(k J ). On the other hand, σ L 2 ()(K J hi) = σ L 2 ()(K J ) c and c > r(k J ), then we have that σ L 2 ()(K J hi) is real and nonpositive. Finally, since the spectrum is independent of X, we obtain the result. ii. Under the hypotheses we have that K L(X, X) is compact, then thanks to Theorem 2.4.4, we know that σ X (K h I) = Im( h ) {µ n } M n=1, with M N or M =. Since Im( h ), then is an isolated eigenvalue with finite multiplicity. If M =, then {µ n } n=1 is a sequence of eigenvalues of K h I with finite multiplicity, that has accumulation points in Im( h). As in part i. we obtain that K J h I is a selfadjoint operator in L 2 (). Then, thanks to Proposition 2.3.1, (K J h )u, u L 2 (),L 2 () = J(x, y)u(y) u(x) dydx J(x, y) u 2 (x)dydx = 1 J(x, y)(u(x) u(y)) 2 dy dx, 2 (2.42) Then from Proposition , and the equality (2.42) we know that σ L 2 ()(K J h ) sup (K J h )u, u L 2 (),L 2 (). (2.43) u L 2 () u L 2 =1 () Thus, the spectrum is nonnegative. Let us prove below that {} is a simple eigenvalue. We consider ϕ an eigenfunction associated to {}. Thanks to Proposition in L 2 () we have = (K h I)(ϕ), ϕ L 2 (),L 2 () = 1 J(x, y)(ϕ(y) ϕ(x)) 2 dy dx. (2.44) 2 Since J(x, y) >, x, y such that d(x, y) < R, then for all x, ϕ(x) = ϕ(y) for any y B R (x). Thus, ϕ is a constant function in. Therefore, we have proved that {} is a simple eigenvalue. iii. Let us see which is the sign of the spectrum of the operator K J hi, with h h. From (2.43), we have (K J hi)u, u L 2 (),L 2 () = (K J h + h h)u, u L 2 () = (K J h )u, u L 2 () + (h h)u, u L 2 (). 42

76 Chapter 3 The linear evolution equation Throughout this chapter, we will assume that (, µ, d) is a metric measure space, with µ as in Definition Let X = L p (), with 1 p or X = C b (). The problem we are going to work with in this chapter, is the following { u t (x, t) = (K hi)(u)(x, t) = L(u)(x, t), x, t > u(x, ) = u (x), x, (3.1) with u X, K = K J L(X, X) and h L () or h C b (). This means that the operator L = K hi L(X, X). We will apply the results of the linear nonlocal diffusive operator K hi developed in the previous chapter to study the existence, uniqueness, positivity, regularizing effects and the asymptotic behavior of the solution of (3.1). In this chapter, we will prove the existence and uniqueness of solution of (3.1) using the semigroup theory. We will write the solution of the problem (3.1) in terms of the group e Lt, associated to the linear and continuous operator L. In fact, in Proposition 3.1.2, we prove that if K hi L(X, X), for an initial data u X there exists a unique strong solution of (3.1), such that u C (R, X). The comparison, positivity and monotonicity results are well-known for the classical diffusion problem with the laplacian (see for example [25]). We prove such results for the nonlocal problem (3.1). In particular, we will prove that under hypothesis (2.3) on the positivity of J, and R-connected (see Definition ), we have a strong maximum principle, i.e., if the initial data u, then the solution to (3.1), u(t), is strictly positive for all t >. One of the main differences between the nonlocal diffusion and the local diffusion problem is that the solution of (3.1) does not have regularizing effects in positive time, since the solution carries the singularities of the initial data. However, we will see that the semigroup S(t) associated to the operator L = K hi can be written as S(t) = S 1 (t) + S 2 (t), where S 1 (t) is the part that is not compact, but it decays to zero exponentially as time goes to infinity; and S 2 (t) is compact. Then S(t) is asymptotically smooth, according to the definition in [32, p. 4]. 43

77 We are also interested in the asymptotic behavior of the solution of (3.1), i.e., in describing the behaviour of the solution when time goes to infinity. We use the Riesz projection, (see [24, chap. VII]), and we prove Theorem which states that if σ X (K hi) is a disjoint union of two closed subsets σ 1 and σ 2 with Re(σ 1 ) δ 1, Re(σ 2 ) δ 2, with δ 2 < δ 1, then the asymptotic behavior of the solution of (3.1) in X is described by the Riesz Projection of K hi corresponding to σ 1. We prove also that the Riesz projection and the Hilbert projection coincide. Furthermore, we apply this result to the particular cases of the nonlocal diffusion problem (3.1) with h constant or h = h = J(, y)dy. In particular, we recover and generalize the result in [18], for X = L p (), with 1 p or X = C b (), whereas in [18], the authors obtain the result with open R N, in X = L 2 () if the initial data is in L 2 (), and in X = L () if the initial data is in C(). 3.1 Existence and uniqueness of solution of (3.1) Let Y be a Banach space. We start this section defining the group associated to a general linear bounded operator F. For more information, see [43] or [36]. If F L(Y, Y ), the operator e F t can be defined by the Taylor series e F t t k F k =, t R (3.2) k! k= which converges for any t. Thus e F t also belongs to L(Y ). It also has the group property e F (s+t) = e F s e F t, for s, t R. We call e F t the group associated to the operator F, and it satisfies that d dt e F t = F e F t = e F t F. Moreover, it is a uniformly continuous group (see [43, p. 2]). Lemma Let Y be a Banach space. If F L(Y, Y ) then the unique solution of the problem { ut = F (u), u() = u Y is given by u(t) = e F t u, that is differentiable in time and the mapping (3.3) R t u(t) = e F t u Y is analytic. Moreover the mapping (t, u ) e F t u is continuous. 44

78 We apply this semigroup technique to prove the existence of solution of the problem (3.1). The following proposition states the uniqueness and existence of strong solution to the problem (3.1).The hypothesis on the linear operator K, if K is an operator with kernel J, can be verified using Proposition 2.1.1, and the hypothesis on hi can be checked using Proposition Proposition Let (, µ, d) be a metric measure space. If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If K L(X, X) then the problem (3.1) has a unique strong solution u C (R, X), given by u(t) = e Lt u, where e Lt L(X, X) is the group associated to the operator L = K hi. Proof. Since L L(X, X), applying Lemma to the problem (3.1), and we obtain the result. We denote the group associated to the operator L = K hi with S K, h, to remark the dependence on K and h. Hence the solution of (3.1) is u(t, u ) = S K, h (t)u = e Lt u. (3.4) Remark Another big difference between the nonlocal problem (3.1) and the local problem with the laplacian is that for the local problem, the flow is not reversible at all, and as a consequence of the previous Proposition 3.1.2, the flow of the nonlocal problem is reversible. 3.2 The solution u of (3.1) is positive if the initial data u is positive We consider the operator K J (u) = J(x, y)u(y)dy, with J nonnegative, then we prove the Weak Maximum Principle, i.e., the solution u of the problem (3.1) with a nonnegative initial data u (x) is nonnegative. First of all, let us consider the problem (3.1), with h, du dt = K J(u), (3.5) u() = u. Formally, if J then u t (x, ) = K J (u )(x), thus u increases with time and then u since u. The rigorous proof of this is that thanks to (3.2) and Lemma 3.1.1, the solution to (3.5) is given by ( ) u(x, t) = e K J t t k KJ k u (x) = u (x). (3.6) k! 45 k=

79 Since J is nonnegative, we have that K k J (u ) is nonnegative for any u nonnegative, k N. Then we have that the solution u(x, t) is nonnegative. In fact, for any m u(x, t) u (x), u(x, t) u (x)+tk J (u )(x), and u(x, t) Now, for h, let u be the solution to (3.1). We take the function v(t) = e h( )t u(t), for t. ( m k= t k K k J k! ) u (x). This function v satisfies that v() = u, and v t (x, t) = h(x)e h(x)t u(x, t) + e h(x)t u t (x, t) = h(x)e h(x)t u(x, t) + e h(x)t( K(u)(x, t) h(x)u(x, t) ) = e h(x)t K(u)(x, t). (3.7) If h in (3.1) is constant in, h(x) = α, x, with α R, then v t (x, t) = e αt K(u)(x, t) = e αt J(x, y)u(y)dy = J(x, y)e αt u(y)dy = K(v)(x, t). Then v(t) = e αt u(t) is a solution to the problem dv dt = K(v), v() = u. (3.8) We have already proved that the solution to (3.8) with nonnegative initial data is nonnegative. Thus, the solution u(x, t) = e αt v(x, t) of (3.1) with h constant, is also nonnegative. We study now the case for h L () nonconstant. Thanks to (3.7), we know that v satisfies then v can be written as v t (x, t) = e h(x)t K(u)(x, t), v(x, t) = u (x) + Moreover u(x, t) = e h(x)t v(x, t), then u(x, t) = e h(x)t u (x) + 46 t t and v(x, ) = u (x) e h(x)s K(u)(x, s)ds. e h(x)(t s) K(u)(x, s)ds. (3.9)

80 Let X = L p (), with 1 p, or X = C b (). For every ω X and T >, we consider the mapping F ω : C([, T ]; X) C([, T ]; X), with F ω (ω)(x, t) = e h(x)t ω (x) + Fix T > and consider the Banach space with the norm t X T = C([, T ]; X) ω = max t T ω(, t) X. e h(x)(t s) K(ω)(x, s)ds. The proof of the following lemma is included for the sake of completeness. It gives us the inequalities to prove that the mapping F ω is a contraction in X T, and it is valid for a general operator K, not only for the integral operator K J. Lemma Let (, µ, d) be a metric measure space. If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If K L(X, X), ω, z X, and ω, z X T = C([, T ]; X), then there exist two constants C 1 and C 2 depending on h and T, such that F ω (ω) F z (z) C 1 (T ) ω z X + C 2 (T ) ω z, (3.1) where C 1 (T ) = e h L () T, C 2 (T ) = CT e h L () T, C 2 : [, ) R is increasing and continuous, and C 2 (T ), as T. Proof. Since K L(X, X), and considering h = h + + h, (h + = max{, h} and h = min{, h}), with h L () or h C b (), then we obtain F ω (ω)(, t) F z (z)(, t) X e h( )t (ω z ) X + e h L () T ω z X + Ce h L () T T = C 1 (T ) ω z X + C 2 (T ) ω z. Taking supremum in [, T ], Thus, the result. t e h L () (t s) K(ω z)(s) X ds max t T ω z X F ω (ω) F z (z) C 1 (T ) ω z X + C 2 (T ) ω z. In the following propositions we will prove that the solution u written as in (3.9) is nonnegative given any nonnegative initial data u. To do this, we will prove that the mapping F ω has a unique fixed point in X T, and we will prove that u is nonnegative using Picard iterations. The proposition is valid for a general positive operator K, (i.e., if z, then K(z) ), in particular for K = K J with J. 47

81 Proposition (Weak Maximum Principle) Let (, µ, d) be a metric measure space. If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If K L(X, X) is a positive operator, then for every u X nonnegative, the solution to the problem (3.1) is nonnegative for all t, and it is nontrivial if u. Proof. Thanks to (3.9), we know that the solution to (3.1) can be written as u(x, t) = e h(x)t u (x, t) + t e h(x)(t s) K(u)(x, s)ds = F u (u)(x, t). (3.11) We choose T small enough such that C 2 (T ) in Lemma satisfies that C 2 (T ) < 1. Hence, by (3.1) we have that F u ( ) is a contraction in X T = C ( [, T ]; X ). We consider the sequence of Picard iterations, u n+1 (x, t) = F u (u n )(x, t) n 1, x, t T. Then the sequence u n converges to u in X T. We take u 1 (x, t) = u (x), then for t u 2 (x, t) = F u (u 1 )(x, t) = e h(x)t u (x) + t e h(x)(t s) K(u )(x)ds (3.12) is nonnegative, because K is a positive operator. Thus u 2 (x, t) for all t. Repeating this argument for all u n, we get that u n (x, t) is nonnegative for every n 1, for t. As u n (x, t) converges to u(x, t) in X T, we have that the solution u(x, t) is nonnegative in X T. We have proved that for some T >, that does not depend on u, the unique solution u of the problem (3.1) with initial data u(x, ) = u (x) is nonnegative for all t [, T ]. If we consider again the same problem with initial data u(, T ), then the solution u(, t) is nonnegative for all t [T, 2T ]. Since (3.1) has a unique solution then we have proved that the solution of (3.1), u(x, t) for all t [, 2T ]. Repeating this argument, we have that the solution of (3.1) is nonnegative t. Since, we have proved that the solution u(, t) to (3.1) is nonnegative, and K is a positive operator, from (3.11), we have that Thus, the result. u(, t) e h( )t u ( ), if u. Remark In the previous proposition we can only prove the positivity forwards on time. (see Corollary to see that it is only positive forwards). We prove below that under the same hypotheses on the positivity of the function J, assumed in Proposition , if the initial data u is nonnegative, not identically zero, then the solution to (3.1), is strictly positive for t >. Theorem (Strong Maximum Principle) Let (, µ, d) be a metric measure space. 48

82 If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If K J L(X, X), and J with J(x, y) > for all x, y, such that d(x, y)<r, for some R > and is R-connected. Then for every u, not identically zero, in X, the solution u(t) of (3.1) is strictly positive, for all t >. Proof. Thanks to Proposition 3.2.2, we know that u, and it is not trivial, for all x, and for all t. We take v(t) = e h( )t u(t), then recalling the definition of the essential support in Definition 2.1.1, we have P (u(t)) = P (v(t)), for all t. From (3.7), we know that v satisfies Integrating (3.13) in [s, t], we obtain v(t) = v(s) + v t (t) = e h( )t K(u(t)), t. (3.13) t s v t (r)dr v(s), for any t s. (3.14) Then P (v(t)) P (v(s)), t s. Moreover, since v(t) = e h( )t u(t) and thanks to (3.14), we obtain e h( )t u(t) e h( )s u(s). Hence, u(t) e h( )(t s) u(s). This implies that P (u(t)) P (u(s)), t s. As a consequence of (3.14), we have that for all D, t ( v D (t) = v D (s) + e h( )r D K(u(r))) dr. (3.15) s Since P (v(t)) P (v(s)) for all t s, and from (3.15), we have that P (u(t)) D = P (v(t)) D P (K(u)(r)) D, for all r [s, t]. (3.16) Moreover, applying Proposition to u(s), we have P (K(u)(r)) P (K(u(s))) P 1 R(u(s)) = x P (u(s)) B(x, R) for all r [s, t]. (3.17) Hence, if we consider the set D = PR 1 (u(s)). From (3.16) and (3.17), we have that P (u(t)) P 1 R(u(s)), for all t > s. (3.18) 49

83 Hence the essential support of the solution at time t, contains the balls of radius R centered at the points in the support of the solution at time s < t. We fix t >, and let C be a compact set, then Proposition implies that exists n N, such that C P n (u ) for all n n. We consider the sequence of times t = t n, t n 1 = t(n 1)/n,..., t j = t j/n,..., t 1 = t/n, t =. Therefore, thanks to (3.18), we have that the essential supports at time t, contains the balls of radius R centered at the points in the essential support at time t n 1, P 1 R (u(t n 1)), which contains the balls of radius R centered at the points in the essential support at time t n 2, then P 2 R (u(t n 2)). Hence repeating this argument, we have P (u(t)) = P (u(t n )) P 1 R (u(t n 1)) P 2 R (u(t n 2))... P n R (u ) C. Thus, we have proved that u(t) is strictly positive for every compact set in, t >. Therefore, u(t) is strictly positive in, for all t >. Corollary Under the assumptions of Theorem 3.2.4, if u, not identically zero, and P (u ), then the solution to (3.1) has to be sign changing in, t <. Proof. We argue by contradiction. Let us assume first that there exists t < such that u(, t ). We take u(, t ) as initial data, then solving forward in time u(, t), for all t t. Hence, we arrive to contradiction, and u(t ) is not identically zero. Secondly, let us assume that there exists t < such that u(, t ), not identically zero. We take u(, t ) as the initial data, then thanks to Theorem 3.2.4, the solution to (3.1), satisfies that u(x, ) <, x. Thus, we arrive to contradiction. Now, we assume that there exists t < such that u(x, t ). Let u(, t ) be the initial data, then thanks to Theorem 3.2.4, the solution to (3.1), satisfies that u(x, ) >, x. Thus, we arrive to contradiction. Therefore, the solution has to be sign changing for all negative times. Remark As a consequence of Theorem 3.2.4, we deduce that the flow associated to the problem (3.1), sends the boundary of the positive cone of X, (see Definition ), to the interior of it, when time moves forward. Furthermore, from Proposition 3.1.2, we obtain that the flow is reversible, but despite of this, from Corollary we have that the flow is not symmetric for time t > and t <. 3.3 Asymptotic regularizing effects Let (, µ, d) be a metric measure space, for K L(X, X), we consider the equation u t (x, t) = K(u)(x, t) h(x)u(x, t), for x. (3.19) We have seen above, in (3.9), that the group associated to this equation with initial data u X can be written as S K,h (t)u (x) = e h(x)t u (x) + 5 t e h(x)(t s) K(u)(x, s)ds. (3.2)

84 In general, the group (3.2) has no regularizing effects. In particular, the solution of (3.5) (i.e. (3.19) with h ) is given by (3.6), then ( ) t k KJ k u(x, t) u (x) = u (x), (3.21) k! and even if K : L p () C b (), we obtain that the right hand side of (3.21) is in C b (), but on the left hand side we have the initial data that is in L p (). Hence, the regularity of u is equal to the regularity of the initial data u. Moreover, the solution to (3.1) with h constant is given by u(x, t) = e h t v(x, t), where v is solution of (3.8), then the regularity of u is equal to the regularity of the initial data u. Hence there is no regularizing effect. However, we will prove that there exists a part of the group, that we call S 2 (t) that is compact, so it somehow regularizes. Moreover, there exists another part of the group that we call S 1 (t) which does not regularize, i.e., it carries the singularities of the initial data, but it decays to zero exponentially as t goes to, if h. Thus, we will have a regularizing effect when t goes to. Then S K,h (t) is asymptotically smooth, according to the definition in [32, p. 4]. Now, we introduce Mazur s Theorem (see [24, p. 416]), which is the key to prove that S 2 (t) is compact. Theorem (Mazur s Theorem) Let X be a Banach space, and let B X be compact. Then co(b) is compact, where co(b) is the convex hull or the convex envelope of the set B (smallest convex set that contains B). Lemma Let C be a compact set in X, let T > and F : [, T ] X be continuous. If F (s) C for all s [, T ], then for a fixed t (, T ], 1 t t k=1 F (s)ds co(c ). Proof. For a continuous function, the integral is given by t F (s)ds = lim F ( t i ) t i, t where t i [t i 1, t i ] belongs to the partition of the interval [, t], t i = t i t i 1 and t is the diameter of the partition.then 1 t F ( t i ) t i = P 1 i j = i = i i t j i F ( t i ) P t i F ( t i ) t i j F ( t i )α i, t j with α i satisfying α i 1, i, and i α i = 1. Moreover F ( t i ) C, then F ( t i )α i co(c ). i 51

85 Therefore, we have that 1 t t F (s)ds co(c ). Proposition Let (, µ, d) be a metric measure space, if X = L p (), with 1 p, we assume h L (), if X = C b (), we assume h C b (). For < t R fixed, we have that the mapping is continuous. M : [, t] X X (s, f) e h( ) (t s) f Proof. Assume we have proved that the mapping g : [, t] L () s e h( ) (t s) is continuous, then we prove that the mapping M is continuous. Given (s 1, f 1 ) [, t] X, for all ε >, there exist δ 1, δ 2 R positive, such that for all (s 2, f 2 ) satisfying s 1 s 2 < δ 1 and f 1 f 2 X < δ 2, we have that M(s 1, f 1 ) M(s 2, f 2 ) X = e h( ) (t s1) f 1 e h( ) (t s2) f 2 X = e h( ) (t s1) f 1 e h( ) (t s2) f 1 + e h( ) (t s2) f 1 e h( ) (t s2) f 2 X g(s 1 ) g(s 2 ) L () f 1 X + e h( ) (t s2) L () f 1 f 2 X g(s 1 ) g(s 2 ) L () f 1 X + sup e h( ) (t s) L () f 1 f 2 X. s 1 s 2 <δ Since g is continuous, we can choose δ 1 such that ε g(s 1 ) g(s 2 ) L () <, if s 1 s 2 < δ 1 2 f 1 X ε and we choose δ 2 < 2 sup e h( ) (t s), then we obtain that for these δ 1 and δ 2 L () B δ (s 1 ) M(s 1, f 1 ) M(s 2, f 2 ) X < ε Hence, we have proved that M is continuous. Now, we just have to prove that the mapping g is continuous. Given s 1 [, t]. Let us prove that for all ε > there exists δ >, such that if s 1 s 2 < δ then g(s 1 ) g(s 2 ) L () < ε, g(s 1 ) g(s 2 ) L () = e h( )(t s1) e h( )(t s2) L () = e h( )(t s 1) ( 1 e h( )(s 2 s 1 ) ) L () e t h L () 1 e h L () s 2 s 1 = C 1 e C 1 s 2 s 1. We know that the exponential function is continuous, then we have that 1 e C 1 s 2 s 1 as s 2 s 1. Therefore we have that g : [, t] L () is continuous. 52

86 In the following proposition, we see that in general, the solution associated to the problem (3.1), u(t) = S K, h (t) = S 1 (t) + S 2 (t), has no regularizing effects. We prove that S 2 (t) is compact, but S 1 (t) is not. However, we prove that if h is strictly positive in, then S 1 (t) decays to zero exponentially as t goes to, so we have a regularizing effect when t goes to, and S K, h (t) is asymptotically smooth. Theorem Let (, µ, d) be a metric measure space, with µ() <. If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). For 1 p q, let X = L q () or C b (). Proposition 2.1.7), and h satisfies If K L(L p (), X) is compact, (see h(x) α > for all x, and u L p (), then the group associated to the problem (3.1), satisfies that with u(t) = S K,h (t)u = S 1 (t)u + S 2 (t)u i. S 1 (t) L(L p ()) t >, and S 1 (t) L(L p (),L p ()) exponentially, as t goes to. ii. S 2 (t) L(L p (), X) is compact, t >. Therefore S K,h (t) is asymptotically smooth. Proof. We write the solution associated to (3.1), as in (3.9), then we have that and we define u(x, t) = S K,h (t)u (x) = e h(x)t u (x) + S 1 (t)u = e h( )t u S 2 (t)u = t e h( )(t s) K(u)(s)ds = t t e h(x)(t s) K(u)(x, s)ds, x e h( )(t s) K(S K,h (s)u )ds. i. Since u L p () and h L () with h α >, then S 1 (t)u = e h( )t u L p () and S 1 (t) u L p () = e h( )t u ( ) L p () e αt u L p (). Therefore S 1 (t) L(L p (),L p ()) e αt, with α >, then it converges exponentially to, as t goes to. ii. Fix t >, as h L (), S K,h (s) L(L p ()) s [, t], and K L(L p (), X), then S 2 (t)(u ) X t e αt K(S K,h (s)u ) X ds e αt t max s t K(S K,h(s)u ) X <. 53

87 Thus S 2 (t) L(L p (), X). Let us see now that S 2 (t) L(L p (), X) is compact t >. Fix t > and consider a bounded set B of initial data. We know that We denote S 2 (t)u = t S 2 (t)u = F u (s)ds, with t e h( )(t s) K(u)(, s)ds. F u (s) = e h( )(t s) K(S K,h (s)u ). Assume we have proved that F u (s) C, were C is a compact set in X, for all s [, t] and for all u B. Then applying Lemma to F u we have that 1 t S 2(t)(u ) co ( C ), u B, and thanks to the Mazur s Theorem 3.3.1, we obtain that 1 t S 2(t)(B) is in a compact set of X. Therefore S 2 (t) is compact. Now, we have to prove that F u (s) = e h( )(t s) K(S K,h (s)u ) belongs to a compact set, for all (s, u ) [, t] B. First of all, we check that K(S K,h (s)u ) belongs to a compact set W in X, for all (s, u ) [, t] B. Since K is compact, we just have to prove that the set B = {S K,h (s)u : (s, u ) [, t] B} is bounded. In fact, we have that K hi L(L p (), L p ()), then σl p ()(K hi) K hi L(L p ()) δ <, thus S K,h (s)u L p () = u(, s) L p () Ce (δ+ε)s u L p () Ce (δ+ε)t u L p (), for all (s, u ) [, t] B. (This inequality will be proved with more details in Proposition 3.4.2). Then, since B is bounded, we obtain that B is bounded in L p (). Finally, we just need to prove that F u (s) is in a compact set for all (s, u ) [, t] B, and this is true thanks to Proposition that says that the mapping M : [, t] X X (s, f) e h (t s) f is continuous. Then M sends the compact set [, t] W into a compact set C. Thus, F u (s) belongs to a compact set, C, (s, u ) [, t] B. Therefore we have finally proved that S 2 (t) is compact in X, for all t >, and S K,h (t) is asymptotically smooth. 3.4 The Riesz projection and asymptotic behavior In this section we want to study the asymptotic behavior of the solution of the problem (3.1). For this, we need first to introduce the concept of Riesz projection of a linear and bounded operator. Moreover, we will prove that the Riesz projection is equivalent to the Hilbert projection in L 2 (). The Riesz projection is given in terms of the spectrum of the 54

88 operator. Since the spectrum of the operator L = K hi has been proved in Proposition to be independent of X = L p (), with 1 p or X = C b (), then the asymptotic behavior of the solution of (3.1) will be characterized with the Riesz projection, and it can we calculated in X with the Hilbert projection. Consider a general operator F L(Y, Y ), where Y is a Banach space. The proposition below gives a bound of the norm of the group associated to the linear and bounded operator F. We will also give a general result of asymptotic behavior of the solutions associated to the problem { u t (x, t) = F (u)(x, t) (3.22) u(x, ) = u (x), with u Y The definitions below, can be found in [24, chap. VII]. Definition Let F L(Y ), and f be an analytic function in some neighborhood of σ(f ) C, and let U be an open set whose boundary Γ consists of a finite number of rectifiable Jordan curves, oriented in the positive sense. Suppose that U σ(f ), and that U Γ is contained in the domain of analyticity of f. Then the operator is well defined and f(f ) L(Y, Y ). f(f ) = 1 2πi Γ f(λ)(λi F ) 1 dλ If F is a continuous operator the eigenvalues of the operator F are bounded, and there exists δ R such that Re(λ) δ for λ σ(f ). We can find a closed rectifiable curve Γ that contains Figure 3.1: Bounded spectrum σ(f ), without crossing any λ σ(l), like the curve Γ in Figure 3.1. In particular, f(λ) = e λt is analytic in a neighborhood of σ(f ). Definition 3.4.1, and we obtain that e F t F k t k = = 1 e λt (λi F ) 1 dλ. k! 2πi Γ k= Thus, we can apply In the next proposition we estimate the norm of the group e F t : Y Y. 55

89 Proposition Let F L(Y ) be an operator as the one described above with Re(σ(F )) δ, then ε > there exists a constant C = C (ε) such that e F t L(Y ) C e (δ+ε)t t. Proof. For every curve Γ that satisfies the hypotheses of Definition 3.4.1, we have that Re(λ) δ + ε, ε >, λ Γ, then for t e F t L(Y ) = 1 e λt (λi F ) 1 dλ 2πi 1 e Re(λ) t (λi F ) 1 d λ C e (δ+ε)t. 2π Γ Γ Corollary Let F L(Y ) be an operator as the one described above with δ Re(σ(F )) δ, then ε > there exists a constant C = C (ε) such that e F t L(Y ) C e (δ+ε) t t R. Now, we introduce the Riesz projection, that will help us study the asymptotic behavior of the solution of (3.22). The following definitions can be found in [31, chap. 1]. Let F be a bounded and linear operator on the Banach space Y. If N is a subspace of Y invariant under F, then F N denotes the restriction of F to N, which has to be considered as an operator from N into N. A set σ 1 is called an isolated part of σ(f ) if both σ 1 and σ 2 = σ(f ) \ σ 1 are closed subsets of σ(f ). Given an isolated part σ 1 of σ(f ) we define Q σ1 to be the bounded linear operator on Y given by Q σ1 = 1 (λi F ) 1 dλ, 2πi Γ where Γ consists of a finite number of rectifiable Jordan curves, oriented in the positive sense around σ 1, separating σ 1 from σ 2. This means that σ 1 belongs to the inner region of Γ, and σ 2 belongs to the outer region of Γ. The operator Q σ1 is called the Riesz projection of F corresponding to the isolated part σ 1, and is independent of the path Γ described as above. The following theorem and corollary describe some properties of the Riesz projection Q σ1 (see [31, p. 1]). Theorem Let σ 1 be an isolated part of σ(f ), and put U = Im Q σ1 and V = Ker Q σ1. Then Y = U V, the spaces U and V are F invariant subspaces and considering F U = F 1 and F V = F 2 σ(f 1 ) = σ 1 σ(f 2 ) = σ(f ) \ σ 1. Corollary Assume σ 1 is an isolated part of σ(f ), and σ 2 = σ(f ) \ σ 1. Then, Q σ1 + Q σ2 = I Q σ1 Q σ2 =. 56

90 The following lemma will be useful to prove the next proposition, that states that the group, e F t, and the Riesz projection, Q σ, commute, with this we will estimate the norm of the Riesz projection of the solutions to (3.22). Lemma Let Y be a Banach space, F : Y Y and λ ρ(f ) C, then, (λi F ) 1 F k = F k (λi F ) 1, for all k N. Proof. Take x Y, such that there exists y Y satisfying (λi F )y = x, this is y = (λi F ) 1 x. Then λy F y = x. Applying F, we have λf y F 2 y = F x. Thus we have proved that (λi F )F y = F x, applying (λi F ) 1, we have that F y = (λi F ) 1 F x. Since y = (λi F ) 1 x, we have proved that F (λi F ) 1 = (λi F ) 1 F. Following this same argument, we can prove that (λi F )F k y = F k x, for any k N. Hence, the result. Proposition Let σ be an isolated part of σ(f ), then, where, F 1 = F Im Q σ. Proof. Let e F t = k= e F t Q σ = Q σ e F t = e F 1t, (F t) k k!, then thanks to Lemma 3.4.6, (Q σ e F t ) = Q σ ( k= k= ) (F t) k k! = 1 ( (λ F ) 1 (F t) k dλ 2πi Γ k! k= = 1 (λ F ) 1 F k t k dλ 2πi k= Γ k! = 1 F k t k (λ F ) 1 dλ 2πi k= Γ k! F k t k 1 = (λ F ) 1 dλ = (e F t Q σ ). k! 2πi Γ ) Let Y be a Banach space, we study now the asymptotic behavior of the solution of { u t (t) = F (u)(t), with t R (3.23) u() = u, with u Y where F L(Y, Y ) and σ(f ) is a disjoint union of two closed subsets σ 1 and σ 2. Assume like in Figure 3.2. δ 2 < Re(σ 1 ) δ 1, Re(σ 2 ) δ 2, with δ 2 < δ 1, 57

91 Applying Corollary 3.4.5, we have that the solution to (3.23), can be written as u(t) = Q σ1 (u)(t) + Q σ2 (u)(t). On the other hand, the solution of (3.23) is equal to u(t) = e F t u. Thus, thanks to Propositions and 3.4.7, and since Re(σ 2 ) δ 2 we obtain that for t > where, F 2 = F Im Q σ2. Q σ2 (u(t)) Y = ( Q σ2 e F t) (u ) Y = e F 2t (Q σ2 (u )) Y C 2 e (δ 2+ε)t Q σ2 (u ) Y, ε >, (3.24) Figure 3.2: σ(f ) = σ 1 σ 2. The following Theorem, which is the principal result of this section. It states which is the asymptotic behavior of the solution associated to (3.23). Theorem Consider F L(Y ) and let σ(f ) be a disjoint union of two closed subsets σ 1 and σ 2, with δ 2 < Re(σ 1 ) δ 1, Re(σ 2 ) δ 2, with δ 2 < δ 1. Then the solution of (3.23) satisfies lim t e µ t( u(t) Q σ1 (u)(t) ) Y =, µ > δ 2. Proof. By using the definition of the Riesz projection, taking µ > δ 2 and thanks to Corollary e µt (u(t) Q σ1 (u)(t)) = e µt Q σ2 (u)(t) Thanks to (3.24), we know that the right hand side of the latter equation satisfies, e µt Q σ2 (u)(t) Y C 2 e ( µ+δ 2+ε)t Q σ2 (u ) Y, ε >, t > Furthermore, there exists ε > such that ε such that < ε < ε, it happens that Hence, the result. ( µ + δ 2 + ε) <. 58

92 In the following proposition we prove that the Hilbert projection over the space generated by the eigenfunction associated to the first eigenvalue of F, is equal to the Riesz projection in X for a general operator F : X X. We denote by λ 1 the largest eigenvalue associated to F in X. We assume that λ 1 is isolated and simple, and Φ 1 is an eigenfunction associated to λ 1, with Φ L 2 () = 1. Taking σ 1 = {λ 1 }, we know that in the Hilbert space L 2 (), P σ1 (u) = u, Φ 1 Φ 1 = u(x) Φ 1 (x)dx Φ 1, u L 2 (), where P σ1 is the Hilbert projection over the space generated by the eigenfunction associated to σ 1. Proposition Let (, µ, d) be a metric measure space, with µ, as in Definition For 1 p < p 1, with 2 [p, p 1 ], let X = L p (), with p [p, p 1 ], or X = C b (). We assume F L(X, X) is selfadjoint in L 2 (), the spectrum of F, σ X (F ), is independent of X, and the largest eigenvalue associated to F, λ 1 is simple and isolated, with associated eigenfunction Φ 1 L p () L p (), for p [p, p 1 ], if X = L p (), or Φ 1 C b () L 1 (), if X = C b (), and Φ L 2 () = 1. If σ 1 = {λ 1 }, and Γ is the curve around λ 1, such that Γ only surrounds {λ 1 }, then for u X, Q σ1 (u) = 1 (λi F ) 1 u dλ = u(x) Φ 1 (x)dx Φ 1 = u, Φ 1 Φ 1 = P σ1 (u), 2πi Γ where Q σ1 is the Riesz projection and P σ1 is the Hilbert projection over the space generated by the eigenfunctions associated to σ 1. Proof. We consider L 2 () = [Φ 1 ] [Φ 1 ]. Let v(λ) be defined as, then we can write v as follows (λi F )v(λ) = u, v(λ) = a(λ)φ 1 + W (λ), (3.25) where W (λ) [Φ 1 ] and a(λ) = v(λ), Φ 1. Now we want to describe Q σ1 (u) Q σ1 (u) = 1 (λi F ) 1 u dλ 2πi Γ = 1 v(λ)dλ 2πi Γ = 1 a(λ)dλ Φ W (λ) dλ. 2πi 2πi Since λ 1 is a simple eigenvalue, we have that Γ Γ (3.26) Q σ1 (u) = α Φ 1, with α = Q σ1 (u), Φ 1. (3.27) Then, multiplying (3.26) by Φ 1 and integrating in, we obtain 1 Q σ1 (u), Φ 1 = a(λ)dλ Φ 2 1 2πi 1dx + 2πi Γ 59 Γ W (λ) dλφ 1 dx (3.28)

93 Since W (λ) [Φ 1 ] and Φ 1 L 2 () = 1, from (3.28) we have that Q σ1 (u), Φ 1 = 1 a(λ)dλ 2πi Therefore, Γ Q σ1 (u) = 1 a(λ) dλ Φ 1. (3.29) 2πi Γ Now, we compute the Hilbert projection of u in L 2 (). We multiply (3.25) by Φ 1 Since F is selfadjoint in L 2 (), (3.3) is equal to λ v(λ), Φ 1 F v(λ), Φ 1 = u, Φ 1. (3.3) λ v(λ), Φ 1 v(λ), F Φ 1 = u, Φ 1. (3.31) Now, since Φ 1 is an eigenfunction associated to λ 1, (3.31) becomes λ v(λ), Φ 1 λ 1 v(λ), Φ 1 = u, Φ 1 (λ λ 1 ) v(λ), Φ 1 = u, Φ 1. (3.32) Thanks to definition (3.25) and (3.32), we obtain that Finally, from (3.29) and (3.33), Q σ1 (u) = 1 a(λ) dλφ 1 = 1 2πi 2πi Γ a(λ) = v(λ), Φ 1 = u, Φ 1 λ λ 1. (3.33) Γ u, Φ 1 λ λ 1 dλ Φ 1 = u, Φ 1 Φ 1 = P σ1 (u). We have proved the equality in the Hilbert space L 2 (), but we want to prove that this is true also in L p () for p [p, p 1 ]. Since the spectrum, σ X (F ), is independent of X, we have that the projection P σ1 (u) = u, Φ 1 Φ 1 is well defined for u X because by hypothesis, Φ 1 L p () L p () for all p [p, p 1 ], if X = L p (), or Φ 1 C b () L 1 (), if X = C b (). On the other hand, we consider the set V = span [ χ D ; D with µ(d) < ], where χ D is the characteristic function of D. Then, from Proposition 2.1.1, we know that V L 2 () is dense in L p (), with 1 p, and L 2 () C b () is dense in C b (). Since P σ1 Q σ1 in L 2 (), then we have that two linear operators are equal in a dense subspace of X, then they are equal in X. Hence, the result. 3.5 Asymptotic behaviour of the solution of the nonlocal diffusion problem Let (, µ, d) be a metric measure space with compact. In this section be apply the results of the previous section about the asymptotic behavior of the solution for the problem { u t (x, t) = (K hi)(u)(x, t), x, t >, (3.34) u(x, ) = u (x), with u X. 6

94 We study two problems to which we apply the results of the previous sections. In particular we are going to study the asymptotic behavior of the solution of (3.34) with: h constant h = h = J(, y)dy, with J L (, L 1 ()). For h constant For h = a constant we have the problem { u t (x, t) = (K ai)(u)(x, t), with a R, u(x, ) = u (x), with u L p (). (3.35) In the following proposition, we prove that the exponential decay in X of the asymptotic behaviour of the solution of (3.35) is given by the first eigenvalue λ 1 of K ai, and the asymptotic behaviour of the solutions is described by the unique eigenfunction, Φ 1, associated to λ 1. Proposition Let (, µ, d) be a metric measure space, with compact and connected. Let X = L p (), with 1 p, or X = C b (). Let K L(L 1 (), C b ()) be compact, (see Proposition to check compactness of integral operators K with kernel J, and assume J(x, y) = J(y, x) with J(x, y) >, x, y such that d(x, y) < R, for some R >. (3.36) Then the solution u of (3.35) satisfies that where C = lim t e λ 1t u(t) C Φ 1 X =, (3.37) u (x)φ 1 (x)dx Φ 1(x) 2 dx, and Φ 1 is the eigenfunction associated to λ 1. Moreover, Φ 1 L p () L p (), and Φ 1 C b () L 1 (). Proof. From Proposition 2.1.2, we have that σ X (K) is independent of X. Moreover, since J(x, y) = J(y, x), then from Proposition , we know that σ(k) \ {} is a real sequence of eigenvalues {µ n } n N of finite multiplicity that converges to. Furthermore, the hypotheses of Proposition are satisfied, then the largest eigenvalue, λ 1 = r(k), is and isolated simple eigenvalue, and the eigenfunction Φ 1 C b () associated to it, is positive. Since the spectrum does not depend on X, we have that, Φ 1 X, in particular Φ 1 L p () L p (), and Φ 1 C b () L 1 (). Thanks to Proposition 2.4.4, we know that the spectrum, σ X (K ai) \ { a}, is a real sequence of eigenvalues {λ n } n N = {µ n } n N a, of finite multiplicity that converges to a. Now, we consider σ 1 = {λ 1 } and σ 2 = {λ 2,..., λ n,...} { a}, and let Φ 1 be a positive eigenfunction associated to λ 1. Since J(x, y) = J(y, x), from Proposition , K ai is selfadjoint in L 2 (), then we can apply Proposition Then, it holds that Q σ1 (u ) = P σ1 (u ) = C Φ 1, (3.38) 61

95 where C = u (x)φ 1 (x)dx Φ 1(x) 2 dx. Furthermore, for u X thanks to Theorem 3.4.8, the solution of (3.35) satisfies lim t e λ 1t ( u(t) Q σ1 (u)(t) ) X =. (3.39) Since u(x, t) = e (K ai)t u (x), P σ1 = Q σ1, and thanks to Proposition we have that P σ1 (u)(x, t) = P σ1 (e (K ai)t u )(x, t) = e (K ai)t P σ1 (u )(x). (3.4) On the other hand, since P σ1 (u )(x) = C Φ 1, where C = u (x)φ 1 (x)dx Φ 1(x) 2 dx, then e (K ai)t P σ1 (u )(x) = e (K ai)t C Φ 1. (3.41) Moreover, Φ 1 is an eigenfunction associated to λ 1 and e (K ai)t = (K ai) n t n n!, then we have e (K ai)t Φ 1 = (K ai) n t n n! Hence, from (3.4), (3.41) and (3.42), Therefore, thanks to (3.39) and (3.43) Φ 1 = (K ai) n Φ 1 t n n! = λ n 1 Φ 1 t n n! = e λ 1t Φ 1. (3.42) P σ1 (u)(x, t) = C e λ 1t Φ 1 (x). (3.43) lim t e λ 1t u(t) C Φ 1 X =. For h = h L () We consider the problem { u t (x, t) = (K h I)(u)(x, t) u(x, ) = u (x), with u L p () (3.44) In the following proposition, we prove that the solution of (3.44) goes exponentially in norm X to the mean value in of the initial data. Proposition Let (, µ, d) be a metric measure space, with µ() <. Let X = L p (), with 1 q or X = C b (), let K L(L 1 (), C b ()) be compact, (see Proposition to check compactness of integral operators with kernel J), and we assume J L (, L 1 ()), J(x, y) = J(y, x) and J(x, y) >, x, y such that d(x, y) < R, for some R >. (3.45) We assume that h (x) > α >, for all x. Then the solution u of (3.44) satisfies that ( lim t e (β 1+ε)t u(t) 1 µ() where β 1 <, and ε > small enough. u (x)dx) X =, (3.46) 62

96 Proof. Since K L(L 1 (), C b ()) is compact, then K L(X, X) is compact. Thanks to Theorem 2.4.4, we know that σ X (K h I) = Im( h ) {µ n } M n=1, with M N or M =. If M =, then {µ n } n=1 is a sequence of eigenvalues of K h I with finite multiplicity, that has accumulation points in Im( h). Moreover, from Proposition 2.4.5, σ X (K h I) is independent of X. From Corollary 2.4.6, we have that σ X (K h I), and is an isolated simple eigenvalue. Moreover, the constant functions v in, satisfy that (K h I)(v) =. Moreover, since J(x, y) = J(y, x) and thanks to Proposition , K h I is selfadjoint in L 2 (), thus, from Proposition , {µ n } R. Hence, we consider σ 1 = {} an isolated part of σ(k h I), with associated eigenfunction Φ 1 = 1/µ() 1/2, and σ 2 = σ(k h I)\{}. Thanks to Proposition 3.4.9, if u X, Q σ1 (u ) = P σ1 (u ) = u Φ 1 dx Φ 1 = = 1 µ() 1 u (x) u (x)dx. µ() 1/2 dx 1 µ() 1/2 (3.47) Thanks to Theorem 3.4.8, the asymptotic behavior of the solution of (3.44) is given by lim e (β1+ε)t (u(t) P σ1 (u)(t)) =, (3.48) X t where β 1 < is upper bound of Re ( σ X (K h I) \ {} ), and ε > small enough, such that β 1 + ε <. We also know that u(x, t) = e (K hi)t u (x), and P σ1 = Q σ1. Then, thanks to Proposition we have that P σ1 (u)(x, t) = P σ1 (e (K h I)t u )(x, t) = e (K h I)t P σ1 (u )(x). (3.49) On the other hand, since Q σ1 (u )(x) = P σ1 (u )(x) = u, Φ 1 Φ 1 = CΦ 1, then e (K h I)t P σ1 (u )(x) = e (K h I)t CΦ 1. (3.5) Furthermore, Φ 1 is an eigenfunction associated to {} and e (K h I)t = (K h I) n t n n!, then we have e (K h I)t Φ 1 = (K h I) n t n n! Hence, from (3.49), (3.5) and (3.51) Φ 1 = (K h I) n Φ 1 t n n! = () n Φ 1 t n n! = e t Φ 1 = Φ 1. (3.51) P σ1 (u)(x, t) = P σ1 (e (K h I)t u )(x, t) = e (K h I)t P σ1 (u )(x) = e t P σ1 (u )(x) = P σ1 (u )(x). (3.52) 63

97 Therefore, thanks to (3.48), (3.52) and (3.47), the asymptotic behavior of the solution of (3.44) is given by ( lim t e (β 1+ε)t u(t) 1 u (x)dx) =. (3.53) µ() X Remark With Propositions and 3.5.2, we recover the result of asymptotic behaviour in [18], but we obtain the results for a general metric measure space instead of an open subset of R N. Moreover we give the asymptotic behaviour in norm X = L p () or X = C b (), whereas in [18] the results are obtained in an open bounded set R N, and the asymptotic behaviour is given in norm L 2 () if u L 2 () and in norm L () if the initial data is in C(). 64

98 Chapter 4 Nonlinear problem with local reaction Throughout this chapter, we will assume that (, µ, d) is a metric measure space, X = L p (), with 1 p, or X = C b (), and the operator K L(X, X). The problem we are going to work with, is the following { u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)) = L(u)(x, t) + f(x, u(x, t)), x, t > u(x, t ) = u (x), x, (4.1) with f : R R representing the local reaction term, and u X. We will write the solution of the problem (4.1), in terms of the group e Lt associated to the linear operator L = K hi. In fact, we will write the solution with the Variation of Constant Formula, (4.6), and we will focus in the study of the existence and uniqueness of the solution associated to (4.1), firstly for f globally Lipschitz and secondly for f locally Lipschitz and satisfying some sign-conditions. If f is globally Lipschitz, we will prove that the solution of (4.1) with initial data u X, is a global strong solution such that u C 1 ([, T ], X) for all T >. We will also give positivity and monotonicity results for the solution, analogous to the results of the local nonlinear reaction-diffusion problem with boundary conditions, (see for example [4]). In particular, we will prove the following monotonicity properties: Given two ordered initial data, the corresponding solutions are ordered. If f(u) for all u. Given a nonnegative data, u, the corresponding solution is nonnegative. If f g. If we denote by u f (t) and u g (t) the solution of (4.1) with nonlinear term f and g respectively. Then u f (t) u g (t). Let u(t) be a supersolution, and let u(t) be the solution. If ū() u() then u(t) u(t) as long as the supersolution exists. inequality. The same is true for subsolutions with reversed 65

99 We will also prove the existence, uniqueness and monotonicity properties of the solution of (4.1) when the nonlinear term f, is locally Lipschitz in the variable s R, uniformly with respect to x, and satisfies sign conditions: there exists C, D R with D >, such that f(x, s)s Cs 2 + D s, for all x. (4.2) After that, we give some asymptotic estimates of the solution, and we will finish proving under hypotheses (4.2) on f, the existence of two extremal equilibria ϕ m and ϕ M in L (). In fact, we prove that all the solutions of (4.1) with bounded initial data will enter between the two extremal equilibria when time goes to infinity for a.e. point in, and if the initial data u is in L p (), with 1 p <, then ϕ M and ϕ m are bounds of any weak limit in L p () of the solution of (4.1), when t goes to infinity. These results are weaker than the results for the local reaction-diffusion equation, where the asymptotic dynamics of the solution enter between the extremal equilibria uniformly in space, for bounded sets of initial data, (see [44]). After studying the asymptotic behaviour we are confined to study the stability of the equilibria of the problem (4.1) with h = h = J(, y)dy. Since F : X X globally Lipschitz is not differentiable (see Appendix B), hence we do not have that if an equilibrium is stable with respect to the linearization, then it is stable in the sense of Lyapunov. We give criterions on f to have similar results, and we prove that any nonconstant equilibria in C b () of (4.1) with h = h is, if it exists, unstable when f is convex. Similar results are obtained in [14, 4] for the local reaction-diffusion problem. 4.1 Existence, uniqueness, positiveness and comparison of solutions with a globally Lipschitz reaction term Let (, µ, d) be a measurable metric space, if X = L p (), with 1 p, we assume h L (), if X = C b (), we assume h C b (), In this section we focus on the existence and uniqueness of solution of the problem { u t (x, t) = L(u)(x, t) + f(x, u(x, t)), x, t > u(x, ) = u (x), x, (4.3) with f globally Lipschitz, whose solution will be denoted as u(x, t, u ). Definition Let X = L p (), with 1 p, or X = C b (), the Nemitcky operator associated to f : R R, is defined as an operator F : X X, such that F (u)(x) = f(x, u(x)), with u X. The following theorem gives a criterium to prove the existence of strong solutions. For more details see [43, p. 19]. 66

100 Theorem Let Y be a Banach space, we assume the linear operator H : Y Y generates a C semigroup in Y, denoted by e Ht. We consider the problem { u t (t) = H(u)(t) + g(t), t > t (4.4) u(t ) = u Y. We assume g C([t, t 1 ], Y ), u D(H) and u is a mild solution of (4.4) given by Moreover, assume either t u(t) = e H(t t) u + e H(t s) g(s)ds. t i. g C([t, t 1 ], D(H)), i.e., t g(t) Y and t Hg(t) Y are continuous, ii. g C 1 ([t, t 1 ], Y ). Then u C 1 ([t, t 1 ], Y ) C([t, t 1 ], D(H)), and it is a strong solution of (4.4) in Y. Let us consider now a general globally Lipschitz operator G : X X, and we study the problem { u t (x, t)=(k hi)(u)(x, t) + G(u)(x, t) = L(u)(x, t) + G(u)(x, t), x, t R (4.5) u(x, ) = u (x), x, In the following proposition we prove the existence and uniqueness of the solution to (4.5). Proposition If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). Let K hi L(X, X) and let G : X X be globally Lipschitz. Then the problem (4.5) has a unique global solution u C((, ), X), for every u X, with u(, t) = e Lt u + t Moreover, u C 1 ((, ), X) is a strong solution in X. e L(t s) G(u)(, s) ds. (4.6) Proof. This proof is standard, however, we give it for the sake of completeness. The solution associated to the equation (4.5) can be written as in (4.6). Denoting by F(u) the right hand side of (4.6), we are lead to look for fixed points of F, in V = C ([ T, T ], X), for some T >. Note that V is a complete metric space for the sup norm. First we prove F maps V into itself. Thus, we prove that for u V, F(u) C([ T, T ], X). First of all, if u V then G(u) C([ T, T ], X), because G : X X is globally Lipschitz. Since L = K hi L(X, X), 67

101 we have that L L(X) < σ(l) < L L(X), then thanks to Corollary 3.4.3, there exists < a, M R such that e Lt L(X) Me a t, for all t R. (4.7) Then, since G(u) C([ T, T ], X) and thanks to (4.7), we have that F(u)(t) X e Lt t u X + e L(t s) G(u)(, s) ds X e Lt t u X + e L(t s) G(u)(s) X ds Me a t t u X + Me a t s G(u)(s) X ds Me a t u X + M t e a t sup s [ t, t ] G(u)(s) X. Thus, we have that F(u)(t) X. To prove continuity in time, we fix t [ T, T ] and ε R, we have that and then F(u)(t + ε) = e Lε F(u)(t)+ t+ε F(u)(t + ε) F(u)(t) X (e Lε I)F(u)(t) X + t e L(t+ε s) G(u)(, s) ds t+ε t e L(t+ε s) L(X) G(u)(s) X ds. The first term on the right hand side above goes to zero when ε goes to zero, because e Lt is a strongly continuous group and F(u)(t) X. In the second term, G(u) C([ T, T ], X), for u V, and e L(t+ε s) L(X) Me a t+ε s, then the integral term is small if ε is small and continuity follows. Thus F(V ) V. Now we prove that F is a contraction on V if T is small enough. If u 1, u 2 V and t [ T, T ], then t F(u 1 )(t) F(u 2 )(t) X e L(t s) L(X) G(u 1 )(, s) G(u 2 )(, s) X ds, since G is globally Lipschitz, and e Lt L(X) Me a t we get F(u 1 )(t) F(u 2 )(t) X t L G e L(t s) L(X) (u 1 (s) u 2 (s)) X ds t ML G e a t s (u 1 (s) u 2 (s)) X ds ML G t e a t sup s [ t, t ] u 1 (s) u 2 (s) X. since t [ T, T ], we have that for T small enough, ML G T e a T < 1. Therefore F is a contraction and has a unique fixed point. Arguing by continuation. Since T does not depend on u, if we consider again the same problem with initial data u(x, T ), then we find that there exists a unique solution for all 68

102 t [, 2T ]. Also, if we consider the same problem with initial data u(x, T ), then we find that there exists a unique solution for all t [ 2T, ]. Thanks to the uniqueness, we have that there exists a unique solution, u, for all t [ 2T, 2T ]. Repeating this argument, we prove that there exists a unique solution, u C 1 ([ T, T ], X), of (4.5) for all T >. We have proved that there exists a unique solution u C([ T, T ], X) of (4.5) T >, that satisfies the Variations of Constants Formula, (4.6). Moreover, consider g(t) = G(u(t)). Since u : [ T, T ] X is continuous, and G : X X is continuous, we have that g : [ T, T ] X is continuous. Moreover, since D(L) = X and L L(X, X), we can apply Theorem Therefore, u C 1 ([ T, T ], X) is a strong solution in X, T >. Now we will prove some monotonicity properties for the problem (4.5). For the linear problem the comparison results were obtained for positive time, (see Corollary 3.2.5), then for the nonlinear problem, (4.5), the results will be also proved for positive time. In the following Proposition we prove that given two initial data ordered, the corresponding solutions remain ordered as long as they exist. Moreover, under the same hypothesis on the positivity of J in Proposition , the solutions are strictly ordered (i.e. u 1 > u 2 ). Proposition (Weak and Strong Maximum Principles) If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). We assume L = K J hi L(X, X), J nonnegative, G : X X globally Lipschitz, and there exists a constant β >, such that G + βi is increasing. (Weak Maximum Principle): If u, u 1 X satisfy that u u 1 then u (t) u 1 (t), for all t, where u i (t) is the solution to (4.5) with initial data u i. (Strong Maximum Principle): In particular if J satisfies that J(x, y)> for all x, y, such that d(x, y)<r, (4.8) for some R >, and is R-connected, then if u u 1, u u 1, u (t) > u 1 (t), for all t >. Proof. We rewrite the equation of the problem (4.5) as u t (x, t) = L(u)(x, t) β u(x, t) + G(u)(x, t) + β u(x, t), where β is the constant in the hypotheses. From Proposition we know that u i (t) is the strong solution of (4.5), with initial data u i, and u i (t) is the unique fixed point of F i (u)(t) = e (L βi)t u i + t e (L βi)(t s) (G(u)(s) + βu(s)) ds (4.9) 69

103 in V = C([ T, T ], X), because F i is a contraction in V provided T small enough for i =, 1. We consider the sequence of Picard iterations, u i n+1(t) = F i (u i n)(t) n 1, for t T. Then the sequence u i n(t) converges to u i (t) in V. Now, we are going to prove that the solutions are ordered for all t [, T ]. We take the first term of the Picard iteration as u i 1 (x, t) = u i(x), then u 1 (t) u1 1 (t), for all t. We also have u i 2(t) = F i (u i 1)(, t) = e (L βi)t u i + t e (L βi)(t s) (G(u i ) + βu i ) ds. Since J is nonnegative, then K J is a positive operator. Moreover, h + β satisfies the same hypotheses as h, and the hypotheses of Proposition are satisfied for L β = K J (h+β)i, then since u u 1, we have that e (L βi)t u e (L βi)t u 1 for all t [, T ]. (4.1) Moreover, since G + βi is increasing and thanks to Proposition 3.2.2, we obtain that e (L βi)(t s) (G(u ) + βu ) e (L βi)(t s) (G(u 1 ) + βu 1 ) for all t [, T ] and s [, t]. (4.11) From (4.1) and (4.11), we have that u 2 (t) u1 2 (t) for all t [, T ]. Repeating this argument, we get that u n(t) u 1 n(t) for all t [, T ], for every n 1. Since u i n(x, t) converges to u i (x, t) in V, we obtain that u (t) u 1 (t) for all t [, T ]. Now, we consider the solution of (4.5) with initial data at time T, u i (T ). Then, since the initial data u (T ) u 1 (T ), are ordered, arguing as above, we obtain that u (t) u 1 (t) for all t [T, 2T ]. Therefore, we have that, u (t) u 1 (t) for all t [, 2T ]. Repeating this argument, we prove that u (t) u 1 (t), for all t. To prove the second part, we know from Proposition that, u i (t), the solution of (4.5) with initial data u i is given by (4.9). Moreover, since h + β and J satisfy the hypotheses of Theorem 3.2.4, we have that e (L βi)t u = e (K (h+β)i)t u > e (K (h+β)i)t u 1 = e (L βi)t u 1, for all t >. And, thanks to the monotonicity of G( ) + βi, we obtain that t e (L βi)(t s) ( G(u )(x, s) + βu (x, s) ) ds for all t. Thus, u (t) > u 1 (t) for all t >. 7 t e (L βi)(t s) ( G(u 1 )(x, s) + βu 1 (x, s) ) ds,

104 In the proposition below, we prove monotonicity properties with respect to the nonlinear term, for the problem (4.5). Proposition If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). If L = K J hi L(X, X), J is nonnegative, G i : X X is globally Lipschitz for i = 1, 2, and there exists a constant β >, such that G i + βi is increasing for i = 1, 2 and then G 1 G 2 u 1 (t) u 2 (t), for all t, where u i (t) is the solution to (4.5) with G = G i and initial data u X. In particular if is R-connected and J satisfies hypothesis (4.8) of Proposition 4.1.4, then u 1 (t) > u 2 (t), for all t >. Proof. Arguing like in previous Proposition 4.1.4, we know that u i (t) is the strong solutions of (4.5) with nonlinear term G i, and u i (t) is the unique fixed point of F i (u)(t) = e (L βi)t u + t e (L βi)(t s) (G i (u)(s) + βu(s)) ds (4.12) in V = C([ T, T ], X), provided T small enough, for i = 1, 2. We have proved in Proposition that F i is a contraction in V provided T small enough. We consider the sequence of Picard iterations, u i n+1(t) = F i (u i n)(t) n 1. Then the sequence u i n(, t) converges to u i (, t) in V. Now, we are going to prove that the solutions are ordered for all t. We take the first term of the Picard iteration as u i 1 (x, t) = u (x), then u i 2(t) = F i (u i 1)(, t) = e (L βi)t u + t e (L βi)(t s) (G i (u ) + βu ) ds. In Proposition we proved that under the hypotheses in this Proposition then we can use Proposition 3.2.2, and thanks to the fact that G 1 + βi G 2 + βi, we have e (L βi)(t s) (G 1 (u ) + βu ) e (L βi)(t s) (G 2 (u ) + βu ), for all t and s [, t]. Hence u 1 2 (t) u2 2 (t) for all t [, T ]. Repeating this argument, we obtain that u 1 n(x, t) u 2 n(x, t) for all t [, T ], for every n 1. Since u i n(t) converges to u i (t), in V, then u 1 (t) u 2 (t) for all t [, T ]. 71

105 Now, we consider the solution of (4.5) with nonlinear term G i and with initial data ũ i (T ) = ui (x, T ), then arguing as above and since the initial data are also ordered, we obtain that ũ 1 (t) ũ 2 (t) for all t [T, 2T ]. Since the solution to (4.5) is unique, then the solutions u i (, t) are ordered for for all t [, 2T ]. Repeating this argument, we obtain that u 1 (t) u 2 (t), for all t. (4.13) To prove the second part, we know from (4.12) that, u i (t), the solution of (4.5) with nonlinear term G i is given by u i (t) = e (L βi)t u + t e (L βi)(t s) ( G i (u i )(s) + βu i (s) ) ds. Thanks to (4.13), and the fact that G 1 + βi is increasing, and G 1 G 2 we have that ( G1 (u 1 )(x, t) + βu 1 (x, t) ) ( G 1 (u 2 )(x, t) + βu 2 (x, s) ) ( G 2 (u 2 )(x, t) + βu 2 (x, t) ), t. (4.14) From (4.14), since h + β L (), and J satisfies the hypotheses of Theorem 3.2.4, we have that e (L βi)t ( G 1 (u 1 )(x, s) + βu 1 (x, s) ) > e (L β)i)t ( G 2 (u 2 )(x, s) + βu 2 (x, s) ), for all t >. (4.15) Therefore, thanks to (4.15), we obtain that t e (L βi)(t s) ( G 1 (u 1 )(x, s) + βu 1 (x, s) ) ds > for all t >. Thus, u 1 (t) > u 2 (t), for all t >. t e (L βi)(t s) ( G 2 (u 2 )(x, s) + βu 2 (x, s) ) ds, The following proposition states that if the initial data is nonnegative, the solution of (4.5) is also nonnegative. Proposition (Weak and Strong Positivity) If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). We assume L = K J hi L(X, X), J nonnegative, G : X X globally Lipchitz, and there exists a constant β >, such that G + βi is increasing, and G(). If u X, with u, not identically zero, then the solution to (4.5), u(t, u ), for all t. In particular if is R-connected and J satisfies hypothesis (4.8) of Proposition 4.1.4, then u(t, u ) >, for all t >. 72

106 Proof. Arguing like in Proposition we know that u(t) is the solution of (4.5), it is strong, and u(t) is the unique fixed point of F(u)(t) = e (L βi)t u + t e (L βi)(t s) (G(u)(, s) + βu(s)) ds (4.16) that is a contraction in V = C([ T, T ], X), provided T small enough. We consider the sequence of Picard iterations, u n+1 (t) = F(u n )(t) n 1, for all t T. Then the sequence u n (, t) converges to u(, t) in V. We take u 1 (x, t) = u (x), the positive initial solution, then u 2 (t) = F(u 1 )(t) = e (L βi)t u + t e (L βi)(t s) (G(u ) + βu ) ds. In Proposition we proved that under the hypotheses in this Proposition then we can use Proposition 3.2.2, then e (L βi)t u, for all t [, T ]. (4.17) Moreover, if G(), β > and G( ) + βi is increasing, then G(u) + βu for all u. Hence, we obtain that e (L βi)(t s) (G(u ) + βu ), for all t [, T ] and s [, t]. (4.18) Hence, from (4.17) and (4.18), u 2 (t) for all t [, T ]. Repeating this argument, we get that u n (, t) is nonnegative for every n 1. Since u n (t) converges to u(t). Thus, the solution u(t) is nonnegative in V, for all t [, T ]. If we consider again the same problem with initial data ũ (t) = u(x, T ), then arguing as above we have that ũ(t) is nonnegative for all t [T, 2T ]. Thanks to the uniqueness of solution we have that u(t) for all t [, 2T ]. Repeating this argument, we prove that the solution of (4.5) is nonnegative t. To prove that the solution u(t) is strictly positive, we know that u(t) is given by (4.16). Moreover, since h + β and J satisfy the hypotheses of Theorem 3.2.4, we have that e (L βi)t u >, for all t >. (4.19) Moreover, since u is nonnegative t, and ( G + βi ) (u) for all u, thanks to Theorem 3.2.4, we have also that t e (L βi)(t s) (G(u)(x, s) + βu(x, s)) ds, for all t. (4.2) Thus, from (4.19) and (4.2), we have that u(t) > for all t >. To prove the following results, we first give the definition of supersolution and subsolution. 73

107 Definition Let X = L p (), with 1 p or X = C b (), we say that u C([a, b], X) is a supersolution to (4.5) in [a, b], if for any t s, with s, t [a, b] u(t) e L(t s) u(s) + We say that u is a subsolution if the reverse inequality holds. t s e L(t r) G(u)(r)dr. (4.21) Remark We assume that e Lt preserves the positivity, i.e., we assume J is nonnegative. If u C([a, b], X) differentiable satisfies that u t (t) L(u)(t) + G(u)(t), for t [a, b] (4.22) then u is a supersolution that satisfies (4.21). The same happens for subsolutions if the reverse inequality holds. Let us prove this below for supersolutions. Since (4.22) is satisfied, there exists f : R X, with f, such that Then u(t) = e Lt u(s) + u t (t) = L(u)(t) + G(u)(t) + f(t), for t [a, b] (4.23) t s e L(t r)( G(u)(r) + f(r) ) dr, for t, s [a, b], s t. (4.24) Since f is nonnegative and e Lt preserves the positivity, then t s el(t r) f(r)dr. Hence, from (4.24) we have that (4.21) is satisfied. Thus, the result. The following proposition states that a supersolution is greater than the solution to (4.5). Proposition If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). Let L = K J hi L(X, X), J be nonnegative, G : X X be globally Lipchitz, and there exists a constant β >, such that G + βi is increasing. Let u(t, u ) be the solution to (4.5) with initial data u X, and let ū(t) be a supersolution to (4.5) in [, T ]. If u() u, then ū(t) u(t, u ), for t [, T ]. The same is true for subsolutions with reversed inequality. Proof. Arguing like in Proposition we know that u(t) is the solution of (4.5), it is strong, and u(t) is the unique fixed point of F(u)(t) = e (L βi)t u + t e (L βi)(t s) (G(u)(, s) + βu(s)) ds (4.25) in C([, τ], X), provided τ small enough. We choose ρ min{τ, T }, then the supersolution ū(t) X exists for all t [, ρ]. Note that ū satisfies by definition that ū(t) F(ū)(t), t [, ρ]. (4.26) 74

108 We consider the sequence of Picard iterations in V = C([, ρ], X), u n+1 (x, t) = F(u n )(x, t) n 1, (4.27) with u 1 (t) = u(t). Then the sequence u n (t) converges to u(t) in V. If we show that, ū u n, a.e. in V, for n = 1, 2, 3,.... (4.28) then, we have the result in V. Since u 1 = ū, then ū u 1 = ū, and (4.28) is satisfied for n = 1. Moreover, thanks to (4.26), we have that ū F(ū) = u 2, then (4.28) is true for n = 2. Assume now for induction ū(t) u n (t), for all t [, ρ]. (4.29) From Proposition we have that F is increasing in V, and thanks to (4.26), (4.27) and (4.29), we have that u F(u) F(u n ) = u n+1, for all t [, ρ]. Then, we have proved (4.28). Moreover, u n (x, t) converges to u(x, t) in V. Then, we have that ū(t) u(t, u ), for all t [, ρ]. Therefore, we have proved that for ρ >, ū(t) u(t, u ), t [, ρ]. Now, we take ρ T, then ū(t) exists for all t [ρ, ρ], with ρ 2τ. If we consider again the same problem with initial data ũ (ρ) = u(, ρ), then ũ(t) is the unique fixed point of F(ũ)(t) = e (L βi)(t ρ) ũ(, ρ) + t ρ e (L βi)(t s)( G(ũ)(, s) + βũ(, s) ) ds in V = C([ρ, ρ], X), and the supersolution satisfies by definition that ū(t) F(ū(t)). Following the same argument as above, we obtain that the supersolution, ū, and the solution, ũ, are ordered for all time t [ρ, ρ]. Thanks to the uniqueness of solution of (4.5) we have that ū(t) u(t, u ), t [, ρ]. With this continuation argument, we prove that the supersolutions are greater or equal to the solution of (4.5), in [, T ]. 4.2 Existence and uniqueness of solutions, with locally Lipchitz f Let (, µ, d) be a metric measure space: If X = L p (), with 1 q, we assume h L (). 75

109 If X = C b (), we assume h C b (). Let K L(X, X), in this section we prove the existence and uniqueness of solution of the problem { u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)) = L(u)(x, t) + f(x, u(x, t)), x (4.3) u(x, t ) = u (x), x, with u X, and f : R R a function that sends (x, s) to f(x, s), that is locally Lipschitz in the variable s R, uniformly with respect to x, i.e., s R, there exists a neighbourhood U of s such that s 1, s 2 U, f(x, s 1 ) f(x, s 2 ) < L U s 1 s 2, x, and f satisfies sign conditions. First of all, we introduce an auxiliary problem associated to (4.3). For k >, let us introduce a globally lipschitz function, f k : R R, associated to the locally Lipschitz function f such that f k (x, u) = f(x, u) for u k, and x. (4.31) Hence, f k is the truncation of the function f. We introduce the following problem, that is equal to (4.3) substituting the locally Lipschitz function f with the associated globally Lipschitz function f k { u t (x, t)=(k hi)(u)(x, t) + f k (x, u(x, t)) =L(u)(x, t) + F k (u)(x, t), x, t R u(x, ) = u (x), x. (4.32) where F k : X X is the Netmitcky operator associated to the globally Lipschitz function f k. The solution of the problem (4.32) will be denoted as u k (t, u ) = S k (t)u. Since the truncation f k is globally Lipschitz, then the associated Nemitcky operator F k is globally Lipschitz (see Appendix B, Lemma ), then we can apply Proposition to obtain the existence and uniqueness of solutions of the problem (4.32). Moreover, since f k is globally Lipschitz, there exists β > such that f k + βi is increasing, then F k + βi is increasing (see Appendix B, Lemma ). Hence, the hypotheses of Propositions 4.1.4, 4.1.5, and are satisfied, and we obtain those comparison results for the problem (4.32). Now, we prove the existence and uniqueness of solutions of (4.3) with initial data u L () or u C b (), under the sign condition (4.33) on the locally lipschitz function f. Proposition Let X = L () or X = C b (). We assume K L(X, X), J nonnegative, and h X, and we assume that the locally lipschitz function f satisfies that there exists a function g C 1 (R), and s, δ > such that (h ( ) h( ))s 2 + f(, s)s g (s)s δ s, s > s, (4.33) 76

110 where h (x) = J(x, y)dy L (), ( J L (, L 1 ()) ). Then there exists a unique global solution of (4.3) with initial data u X, such that u(, t) is given by u(, t) = e Lt u + t Moreover, u C 1 ([, ), X) is a strong solution in X, and e L(t s) f(, u(, s)) ds. (4.34) u(t, u ) L () max { s, u L ()} for all t. (4.35) Proof. Fix M > s. We introduce the auxiliary problem { ż(t) = g (z(t)) z() = M. (4.36) Since g C 1 (R), thanks to Peano s and Picard-Lindelöf Theorems, we have that there exists a unique local solution to (4.36). Thanks to second inequality in (4.33), with a continuation argument we have that z is defined for t. In fact, from (4.33), and since ż(t) = g (z(t)), then z(t) decreases for every t such that z(t) > s, and z(t) > s for all t. Since z() = M, and M > s we have that z(t) M, t. (4.37) We consider a truncated globally Lipschitz function f k, (4.31), associated to f. Let u k (, t, u ) be the solution of (4.32) with initial data u X, such that u L () M. (4.38) Thanks to Proposition we know that there exists a unique strong solution u k (, t, u ) C 1 (R, X) that is given by the Variation of Constants Formula, u k (, t) = e Lt u + t e L(t s) F k (u k )(, s) ds. (4.39) We choose k = M, (4.4) then thanks to (4.37), we have that f k (x, z(t)) = f(x, z(t)), t, x. (4.41) Moreover, since f satisfies (4.33), and from (4.41), we have that f k satisfies (h ( ) h( ))z(t) 2 +f k (, z(t))z(t) g (z(t))z(t) δ z(t), t such that z(t) > s. (4.42) Now, we are going to prove that z is a supersolution of (4.32). Since z is continuous and z() > s, we define t z := inf{t > : z(t) = s }. 77

111 We have that z(t) is independent of the variable x, then K(z(t)) = h z(t). Thanks to (4.42) and the fact that z(t) > s for every t [, t z ), we have that for all t such that < t < t z, we have K(z)(t) hz(t) + f k (, z(t)) = (h h)z(t) + f k (, z(t)) g (z(t)) = ż(t). for all t [, t ]. Hence, we have proved that z is a supersolution of (4.32) in [, t ]. Analogously, let us consider the auxiliary problem { ẇ(t) = g (w(t)) w() = M. (4.43) Arguing as before, we obtain that there exists t w such that for all t < t w, w is a subsolution of (4.32) in [, t ], and w(t) M, t. (4.44) We choose T < min{ t z, t w }, since the initial data u X, satisfies that u L () < M, and M > s, then z(t) and w(t) are subsolution and supersolution, respectively, of (4.32) in [, T ]. Therefore from Proposition 4.1.9, we obtain that w(t, M) u k (t, u ) z(t, M), t [, T ]. (4.45) Moreover, thanks to (4.37), (4.44) and (4.45), we have that u k (t, u ) M = k for all t [, T ]. (4.46) Thanks to (4.38) and since M is fixed at the beginning as M > s, we have that M > max{s, u L ()}. Thanks to the definition of f k, (see (4.31)), and thanks to (4.46), we have that f k (, u k (t)) = f(, u k (t)). Therefore, u k (, t, u ) is a solution of (4.3). Hence, we denote u k (, t, u ) = u(, t, u ), and we have proved the existence of solution of (4.3) for all t [, T ], moreover, u is a strong solution of (4.3) in X, given by (4.34), with u C 1 ([, T ], X), and thanks to (4.46), we have that u(, t, u ) M = k for all t [, T ]. (4.47) In fact (4.47) is satisfied for M = max{s, u L ()}. (4.48) Arguing by continuation, we consider again the same problem (4.3) with initial data ũ (T ) = u(, T, u ), then from (4.47), the initial data is bounded by M, then arguing like we have done before, considering the auxiliary problems (4.36) and (4.43), with M as in (4.48), we will have that there exists an strong solution of (4.3), ũ C 1 ([T, 2T ], X). Since the solution constructed by truncation is unique, then we have proved that there exists an strong solution of (4.3), u C 1 ([, 2T ], X), given by (4.34), and u(, t, u ) M = k for all t [, 2T ]. (4.49) 78

112 Repeating this argument, we prove that for any T >, there exists a strong solution of (4.3) u C 1 ([, T ], X), it is given by the Variation of Constants Formula (4.34), and it satisfies (4.35). Now let us prove the uniqueness of solution. We consider a solution u C([, T ], X), of the problem (4.3) with initial data u X that satisfies (4.34). Since u C([, T ], X), then sup t [,T ] x sup u(x, t, u ) < C. Thus, if we choose k > C, then f k (, u(, t)) = f(, u(, t)) and then the solutions u k of (4.32), is a solution of (4.3). Hence u and u k coincide. Furthermore from Proposition 4.1.3, we have that the solution u k C 1 ([, T ], X) is unique, it is strong and it is given by the Variation of Constant Formula. Thus, we have the uniqueness of the solution of (4.3). In the following proposition we prove existence and uniqueness of solution of (4.3) with initial data bounded, but now, we assume that f satisfies the sign condition (4.5). If C is negative in hypothesis (4.5), then the sign condition (4.5), would imply the sign condition (4.33) with h h, in the previous Proposition Hence in the proposition below, we assume that C >. Proposition Let X = L () or X = C b (). We assume K L(X, X), h, h X, and the locally lipschitz function f satisfies that there exist C, D R, with C > and D such that f(, s)s Cs 2 + D s, s. (4.5) Then there exists a unique solution of (4.3) with initial data u X, such that u(, t) in C([, T ], X), for all T >, with u(, t) = e Lt u + Moreover, we have that u is a strong solution of (4.3) in X. t e L(t s) f(, u(, s)) ds. (4.51) Proof. First of all, let us prove that (h h)s + f(, s) satisfies the hypothesis (4.5). Since f satisfies (4.5) and h, h X, then (h h)s 2 + f(, s)s (h h)s 2 + Cs 2 + D s ( h h L () + C ) s 2 + D s C 1 s 2 + D s. (4.52) We denote C 1 = C to simplify the notation. Fix < M R. We introduce the auxiliary problem { ż(t) = Cz(t) + D z() = M. (4.53) Then the solution of (4.53) is given by z(t) = D C + ect C 2, t R, with C 2 = M + D C, (4.54) 79

113 and z(t) increases for all t R. Let T > be an arbitrary time, then z(t) z(t ) t [, T ]. (4.55) We consider a truncated globally Lipschitz function f k associated to f. We denote by u k (, t, u ) the solution of (4.32) with initial data u X, u X M. Thanks to Proposition we know that there exists a unique strong solution u k (, t, u ) C 1 (R, X) that satisfies the Variation of Constants Formula, u k (, t) = e Lt u + t Given T > and M >, from (4.55) we choose k z(t ). Thanks to the definition of f k, (4.31), and (4.55) we have that e L(t s) F k (u k )(, s) ds. (4.56) f k (, z(t)) = f(, z(t)), t [, T ] (4.57) We prove below that z is a supersolution of (4.32) for every t [, T ]. Since z(t) is nonnegative for all t [, T ], then, thanks to (4.57) and (4.52), and since z(t) is independent of the variable x, we have that K(z(t)) = h z(t). Thus, K(z)(t) hz(t) + f k (, z(t)) = h z(t) hz(t) + f k (, z(t)) Cz(t) + D = ż(t), for all t [, T ], hence, z is a supersolution of (4.32) for every t [, T ]. Let us consider now the auxiliary problem { ẇ(t) = Cw(t) D w() = M, (4.58) Then w(t) = z(t), and we obtain that w(t) < z(t ) t [, T ]. (4.59) Arguing as before, since w(t) is nonpositive for all t [, T ]. Thanks to (4.59), and since w(t) is independent of the variable x, we have that K(w(t)) = h w(t). Thus, K(w)(t) hw(t) + f k (, w(t)) = h w(t) hw(t) + f k (, w(t)) Cw(t) D = ẇ(t). Thus, w is a subsolution of (4.32) for every t [, T ]. Since k z(t ) and u X M, then z(t) and w(t) are supersolution and subsolution of (4.32) in [, T ], respectively. Therefore, from Proposition 4.1.9, we obtain w(t, M) u k (t, u ) z(t, M), t [, T ]. (4.6) 8

114 Thanks to (4.55), (4.59), and (4.6) we have that u k (t, u ) z(t ) k for all t [, T ]. Thanks to the definition of f k, (4.31), we obtain that f k (, u k (t, u )) = f(, u k (t, u )). Thus, u k (x, t, u ) is a solution to (4.3). Hence we denote u k (t, u ) = u(t, u ), and we have proved the existence of solution of (4.3) for all t [, T ], moreover u is a strong solution of (4.3) in X, given by the Variation of Constants Formula (4.51). Therefore, given any M > and any T >, choosing k z(t ), then we have proved the existence of solution of (4.3) with initial data u X M for all t [, T ]. Now, let us prove the uniqueness, arguing like in Proposition 4.2.1, let us consider a solution u C([, T ], X), of the problem 4.3 with initial data u X, given by (4.51). Since u C([, T ], X), then sup t [,T ] x sup u(x, t, u ) < C. Thus, if we choose k > C, then f k (, u(, t)) = f(, u(, t)) and then the solutions u k of (4.32), is a solution of (4.3). Hence u and u k coincide. Furthermore from Proposition 4.1.3, we have that the solution u k C 1 ([, T ], X) is unique, it is strong and it is given by the Variation of Constant Formula. Thus, we have the uniqueness of the solution of (4.3). Remark By using Kaplan s technique, we prove that the hypothesis (4.5) on f in the previous Proposition is somehow optimal, in the sense that if f(, s) = s p with p > 1, then we do not have global existence of the solution of (4.3). Let us consider the nonlinear term f(s) = s p, with p > 1. Let X = L (), we assume K L(X, X), h L (), J(x, y) = J(y, x), and we consider the problem u t = (K hi)u + f(u) = J(, y)u(y)dy h( )u + u p (4.61) u() = u with u L (), and u. Let Φ > be an eigenfunction associated to the first eigenvalue λ 1 of the operator K hi, then (K hi)φ = λ 1 Φ. We set z(t) = u(t)φ. Let us see what equation does z satisfy, dz dt (t) = = u t (x, t)φ(x)dx J(x, y)φ(x)dx u(y, t)dy h(x)φ(x)u(x, t)dx + u p (x)φ(x)dx (4.62) Relabeling variables in the first term of the right hand side of (4.62), since J(x, y) = J(y, x) and Φ be an eigenfunction associated to the first eigenvalue λ 1 of the operator K hi we have 81

115 that dz dt (t) = = J(x, y)φ(y)dy u(x, t)dx λ 1 Φ(x)u(x, t)dx + u p (x, t)φ(x)dx = λ 1 z(t) + u p (x, t)φ(x)dx h(x)φ(x)u(x, t)dx + u p (x, t)φ(x)dx (4.63) Therefore, if we consider that Φ(x)dx is a measure and we denote it by dµ, then we have dz dt (t) = λ 1z(t) + u p (x, t)dµ (4.64) Thanks to Jensen s Theorem (see [46, p. 62]), we know that if µ is a positive measure on a σ-algebra M in a set such that µ() = 1, and g is a convex function, then ( ) g f dµ g f dµ. In this case g(s) = s p with p > 1 is convex, and if we take an eigenfunction Φ such that Φ(x)dx = 1, then from (4.64) and as a consequence of Jensen s Theorem dz dt (t) = λ 1z(t) + u p (x, t)dµ ( ) p (4.65) λ 1 z(t) + u(x, t)φ(x)dx = λ 1 z(t) + z p (t) = F (z(t)). In this case, for p > 1, we have that if z() >> 1, 1 F (z) =. Thus, we do not have global existence of solution of (4.61) for all time in t. Remark In [3], the authors establish that the Fujita exponent coincides with the classical one when the diffusion is given by the Laplacian. In the previous Proposition 4.2.2, we have proved that the solution u of the problem (4.3), with initial data u in L () or in C b () is in fact the solution of the problem (4.32), with a truncated globally Lipschitz function f k associated to f. Then the solution u of (4.3) satisfies all the monotonicity properties that we have proved for the problem (4.32). We enumerate them in the following corollaries. Corollary (Weak and Strong Maximum Principles) Let X = L () or X = C b (). We assume J nonnegative, K L(X, X), h X, and the locally lipschitz function f satisfies that there exist C, D R, with C >, D such that If u, u 1 X, satisfy that u u 1 then f(, s)s Cs 2 + D s, s. (4.66) u (t) u 1 (t), for all t >, 82

116 where u i (t) is the solution to (4.3) with initial data u i. In particular if J satisfies that J(x, y)> for all x, y, such that d(x, y)<r, (4.67) for some R >, and is R-connected, (see Definition ) then u (t) > u 1 (t), for all t >. Corollary Under the hypotheses of Corollary If the initial data u X, and the nonlinear terms f 1, f 2 satisfy f 1 f 2 then u 1 (t) u 2 (t), for all t, where u i (, t) is the solution to (4.3) with nonlinear term f i. In particular if J satisfies hypothesis (4.67) of Corollary 4.2.5, and is R-connected then u 1 (t) > u 2 (t), for all t >. Corollary (Weak and Strong Positivity) Under the hypotheses of Corollary Let f(), if u X, with u, not identically zero, then the solution to (4.3), u(t, u ), for all t. In particular, if J satisfies hypothesis (4.67) of Corollary 4.2.5, and is R-connected then u(t, u ) >, for all t >. Corollary Under the hypotheses of corollary Let u(t, u ) be a solution to (4.3) with initial data u X, and let ū(t) be a supersolution to (4.3) in [, T ]. If ū() u, then ū(t) u(t, u ), for all t [, T ]. The same is true for subsolutions with reversed inequality. In the previous Proposition 4.2.2, we have proved the existence and uniqueness of solution of the problem (4.3) with initial data in L () or in C b (). Now we prove the existence and uniqueness for the problem with initial data in L p () for all 1 < p <, and we prove also that the solution is a strong solution in L 1 (). Theorem Let µ() <, we assume J(x, y) = J(y, x), and the locally Lipschitz function f satisfies that f(, ) L (), and and for some 1 < p < f u (, u) β( ) L () (4.68) f (, u) u C(1 + u p 1 ), (4.69) 83

117 if K L(L p (), L p ()) and h, h L (), then the equation (4.3) with initial data u L p () has a unique global solution given by the Variation of Constants Formula with u(, t) = e Lt u + and it is a strong solution in L 1 (). t e L(t s) f(, u(, s)) ds, (4.7) u C ( [, T ], L p () ) C 1( [, T ], L 1 () ), T >, Proof. We prove that f satisfies the hypothesis in Proposition 4.2.2, i.e., there exists C, D R, with C, D > such that f(, s)s Cs 2 + D s, s R. Let s > be arbitrary. Integrating (4.68) in [, s], we have s f (, t)dt t f(, s) f(, ) s β( )s. β( )dt (4.71) Multiplying (4.71) by s >, and since β, f(, ) L (), we obtain f(, s)s β( )s 2 + f(, )s β L ()s 2 + f(, ) L ()s Cs 2 + D s. Let s < be arbitrary. Integrating (4.68) in [s, ], f (, t)dt s t f(, ) f(, s) s β( )dt β( )s. (4.72) Multiplying (4.72) by s, since s < and β, f(, ) L (), then f(, s)s β( )s 2 + f(, )s β L ()s 2 + f(, ) L ()s Cs 2 + D s. Thus, we have that f satisfies the hypotheses of Proposition 4.2.2, and we have the existence and uniqueness of solutions for (4.3) with initial data u L (). Since L () is dense in L p (), we consider a sequence of initial data {u n } n N L () such that u n u in L p () as n goes to. Thanks to Proposition 4.2.2, we know that the solution of (4.3) associated to the initial data u n L (), satisfies u n t (x, t) = (K hi)(u n )(x, t) + f(x, u n (x, t)) = L(u n )(x, t) + f(x, u n (x, t)). We want to see first that {u n } n N C([, ), L p ()) is a Cauchy sequence in compact sets of [, ). Then we consider u k t (t) u j t (t) = L(uk u j )(t) + f(, u k (t)) f(, u j (t)). (4.73) 84

118 Multiplying (4.73) by u k u j p 2 (u k u j )(t), and integrating in, we obtain 1 d p d t uk (t) u j (t) p L p () = L(u k u j )(t), u k u j p 2 (u k u j )(t) ( ) + f(, u k (t)) f(, u j (t)) u k u j p 2 (u k u j )(t). If we denote u k (t) u j (t) = w(t) and g(w) = w p 2 w L p (), (4.74) then the first term on the right hand side of (4.74) can be divided in two parts as follows. First of all, we write L(w(t)) = K(w(t)) h ( )w(t) + h ( )w(t) h( )w(t). (4.75) Since J(x, y) = J(y, x), K L(L p (), L p ()), and thanks to Proposition 2.3.1, (K h I)(w)g(w) dx = (K h I)(w) w p 2 w dx = 1 J(x, y)(w(y) w(x))(g(w)(y) g(w)(x))dy dx. 2 (4.76) From (4.76), since J is nonnegative and g(w) = w p 2 w is increasing, then h I)(w) w (K p 2 w dx = 1 J(x, y)(w(y) w(x))(g(w)(y) g(w)(x))dy dx. 2 (4.77) Moreover, h, h L (), then the second part of (4.75) applied to (4.74) satisfies (h (x) h(x)) w p (x) dx C w p L p (). (4.78) On the other hand, thanks to the hypothesis (4.68) and the mean value Theorem, there exists ξ = ξ(x, t), such that, the second term on the right hand side of (4.74) satisfies that ( ) f(, u k (t)) f(, u j (t)) u k u j p 2 (u k u j f )(t) = (, ξ) w p u (4.79) β L () w p L p (). Finally, thanks to (4.74), (4.77), (4.78), and (4.79), we obtain Thanks to Gronwall s inequality, d d t uk (t) u j (t) p L p () C uk (t) u j (t) p L p () and taking supremums in [, T ] in (4.8), we get u k (t) u j (t) p L p () ect u k u j p L p (), (4.8) sup u k (t) u j (t) p L p () C(T ) uk u j p L p () (4.81) t [,T ] 85

119 The right hand side of (4.81) goes to zero as k and j go to. Therefore we have that {u n } n C([, ), L p ()) is a Cauchy sequence in compact sets of [, ), and there exists the limit of the sequence {u n } n in C([, T ], L p ()), T >, denoted by u(t) = lim n un (t) and it is independent of the sequence {u n } n. Let us see this below. We choose two different sequences {u n } n and {v n} n that converge to u, and we construct a new sequence {w n} n, that consists of w 2n+1 = u n and w2n = v n, for all n N. Then, wn converges to u. Since the sequence of solutions {w n (t)} of (4.3) associated to the initial values w n is a Cauchy sequence, then there exists a unique limit w(t) = lim n wn (t), and this limit is the same limit of the sequences {u n (t)} n and {v n (t)} n. Thus, the limit is independent of the sequence {u n } n. Let us prove now that the limit u is given by the Variation of Constants Formula (4.7). We integrate (4.69) in [, s], then s f s (, t) s dt C(1 + t p 1 )dt Therefore, we have that Thus, we have proved f(, s) f(, ) C(s + 1 p s p 1 s) f(, s) C(s + 1 p s p ) + f(, ) C(1 + s + s p ) f(, u) C(1 + u + u p ), (4.82) then, since µ() <, and thanks to (4.82), we have that f : L p () L 1 (). Now we prove that f : L p () L 1 () is Lipschitz in bounded sets of L p (). Consider u, v L p () with u L p (), v L p () < M, and < M R, thanks to the Mean Value Theorem, there exists ξ L p (), ξ(x) = θ(x)u(x) + (1 θ(x))v(x), for a.e. x with θ(x) 1 for a.e. x, and ξ L p () < 2M, such that f(u) f(v) = f u (ξ) u v. From hypothesis (4.69) and Hölder s inequality, we have that f(u) f(v) = f u (ξ) u v C ( 1 + ξ p 1) u v ( ( ( Cµ() + C ( 1 + ξ p 1) ) 1/p p u v L p () ξ dx) ) 1/p p u v L p () ( Cµ() + ( u L p () + v L p () C(M) u v L p (), 86 ) p 1 ) u v L p ()

120 Then, since u(t) = lim n un (t) in C([, T ], L p ()), T >, we have that f(u n ) f(u) in C([, T ], L 1 ()) T >. (4.83) Since L L(L 1 (), L 1 ()), then there exists δ >, such that Re(σ(L)) δ. Hence thanks to Lemma 3.4.2, we know that e Lt L(L 1 ()) C e δt. Thus t e L(t s) f(, u n (s))ds t t t t e L(t s) f(, u(s))ds L 1 () e L(t s) (f(, u n (s)) f(, u(s))) ds L 1 () e L(t s) L (f(, 1 () un (s)) f(, u(s))) L 1 () ds e δs f(, u n (s)) f(, u(s))) L 1 () ds Taking supremums in t [, T ] in (4.84), and from (4.83) t e L(t s) f(, u n (s))ds t e L(t s) f(, u(s))ds in C([, T ], L 1 ()), T >. (4.84) Let u L p () be the limit of the sequence {u n } n N, we have already proved that u n u in C ([, T ], L p ()), T >, and since e Lt L(L p ()) C e δt, we obtain that e Lt u n e Lt u in C ([, T ], L p ()) T >. Moreover, since t e L(t s) f(, u n (s))ds = u n (t) e Lt u n and, u n (t) e Lt u n converges to u(t) elt u in C ([, T ], L p ()) T >, as n, and we have that Then we have that t t e L(t s) f(, u(s))ds = u(t) e Lt u. (4.85) e L(t s) f(, u n (s))ds t e L(t s) f(, u(s))ds converges in C ([, T ], L p ()), T >. Hence, we have proved the global existence of the mild solution, u in C ([, T ], L p ()) for all T > of the problem (4.3), because u satisfies that u(t) = e Lt u + t e L(t s) f(, u(s))ds. Moreover, consider g(t) = f(, u(t)). Since u : [, T ] L p () is continuous, and f : L p () L 1 () is continuous, we have that g : [, T ] L 1 () is continuous. Moreover, L L(L 1 (), L 1 ()). Then, by using Theorem for the problem (4.3), we have that the initial data u L p () D(L) = L 1 () and X = L 1 (), thus u 87

121 C([, T ], L p ()) C 1 ([, T ], L 1 ()) and it is a strong solution in L 1 (). Finally, let us prove the uniqueness of the solution of (4.3) with initial data u L p (), such that u C ([, T ], L p ()) C 1 ([, T ], L 1 ()), T >, is a strong solution of (4.3) and the solution is given by the Variations of Constants Formula (4.7). We consider that there exists two different strong solutions u and v. If we follow the steps of this proof from (4.73) to (4.8), replacing u k for u and u j for v, we obtain Since u() = v() = u, then u(t) v(t) p L p () ect u() v() p L p (). (4.86) u(t) v(t) p L p () t. (4.87) Therefore u(x, t) = v(x, t) for a.e. x and t. Thus, the result. Remark In the previous Theorem 4.2.9, the sign condition on f, (4.69), we have not included the case p = 1. This is because if p = 1 in hypothesis (4.69), then we have that f u (, u) C, then f is globally Lipschitz, and we have proved in Proposition 4.1.3, that if f is globally Lipschitz, then we have existence and uniqueness of solution of (4.3) for any initial data in u L q (), with 1 q. In the following Corollaries we enumerate the monotonicity properties that are satisfied for the solution of (4.3) with initial data u L p () with p as in Theorem We apply Corollaries to 4.2.8, that state the monotonicity properties of the solution of (4.3) with initial data bounded. Corollary Let (, µ, d) be a metric measure space, with µ() <, for 1 q p, we assume that K L(L q (), L q ()), and h L (). If the locally Lipschitz function f satisfies that f(, ) L (), and and, for some 1 < p < If u, u 1 L p () satisfy that u u 1 then f u (, u) β( ) L () f (, u) u C(1 + u p 1 ). u (t) u 1 (t), for all t, where u i (t) is the solution to (4.3) with initial data u i. In particular if J satisfies that J(x, y)> for all x, y, such that d(x, y)<r, (4.88) for some R >, and is R-connected, (see Definition ), then u (t) > u 1 (t), for all t >. 88

122 Proof. Given u, u 1 L p (), with u u 1. Since L () is dense in L p () with 1 < p <, then we choose two sequences {u n } n N and {u n 1 } n N in L () that converge to the initial data u and u 1 respectively,and such that u n u n 1, n N. Thanks to Corollary 4.2.5, we know that the associated solutions satisfy u n(t) u 1 n(t), for all t, n N. From Theorem 4.2.9, we know that u i n(t) converges to u i (t), for i =, 1 in C([, T ], L p ()). Therefore u (t) u 1 (t), for all t. Analogously we arrive to u (t) > u 1 (t), for all t >. Corollary Let (, µ, d) be a metric measure space, with µ() <, for 1 q p, we assume that K L(L q (), L q ()), h L (), and the locally Lipschitz functions f 1 and f 2 satisfy that, f i (, ) L (), and for some 1 < p < If the initial data u L p () and then f i u (, u) βi ( ) L () f i (, u) u C(1 + u p 1 ). f 1 f 2, u 1 (t) u 2 (t), for all t, where u i (, t) is the solution to (4.3) with nonlinear term f i and initial data u. In particular if J satisfies hypothesis (4.88) of Corollary , and is R-connected then u 1 (t) > u 2 (t), for all t >. Corollary Under the hypotheses of Corollary Let f(, ), if u L p (), with u, not identically zero, then the solution to (4.3), u(t, u ), for all t >. In particular if J satisfies hypothesis (4.88) of Corollary , and is R-connected then u(t, u ) >, for all t >. Corollary Under the hypotheses of corollary , let u L p (), and let ū(t) be a supersolution to (4.3) in [, T ], (see 4.21), and let u(t, u ) be the solution to (4.3) with initial data u L p (). If ū() u, then ū(t) u(t, u ), for all t [, T ]. The same is true for subsolutions with reversed inequality. 89

123 Under the hypotheses of Propositions 4.1.3, and Theorem we define the nonlinear semigroup associated to (4.1) written as S(t)u = u(t, u ) = e Lt u Asymptotic estimates Let (, µ, d) be a metric measure space. t If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). e L(t s) f(, u(s))ds. Let K L(X, X). Now, we study asymptotic estimates of the norm X of the solution u of the nonlinear nonlocal problem that we recall is given by { u t (x, t) = (K hi)(u)(x, t) + f(x, u(x, t)) = L(u)(x, t) + f(x, u(x, t)), x (4.89) u(x, ) = u (x), x, with u X and f : R R as in Propositions 4.1.3, and Theorem 4.2.9, where the nonlinear term f satisfies that there exist C( ) L () and < D( ) L () such that This means that f(, u)u C( )u 2 + D( ) u (4.9) f(x, u) C(x)u + D(x), if u f(x, u) C(x)u D(x), if u. (4.91) In the following proposition we give more details about C and D, and we give bounds of u(t), where u is the solution to (4.89). Proposition If X = L p (), with 1 p, we assume h L (). If X = C b (), we assume h C b (). Let K L(X, X), and J be nonnegative. We assume either: i. u X, f : R R globally Lipschitz, and f(, ) L (). Then there exists C = L f and D = f(, ) L (), such that f(, u)u C( )u 2 + D( ) u, u. (4.92) ii. u in X = L () or X = C b (), f(x, s) is locally Lipschitz in the variable s R, uniformly respect to x, and there exist C( ), D( ) L () with D such that f(, u)u C( )u 2 + D( ) u, u. (4.93) 9

124 iii. J(x, y) = J(y, x), f is locally Lipschitz in the variable s R, uniformly respect to x, and it satisfies that f(, ) L (), and for some 1 < p <, f u (, u) β( ) L () (4.94) f (, u) u C 1(1 + u p 1 ), (4.95) initial data u in X = L p () and K L(L q (), L q ()), for 1 q p. Then there exists C = β L () and D = f(, ) L (), such that f(, u)u C( )u 2 + D( ) u, u. (4.96) Let U(t) be the solution of { U t (x, t) = L(U(x, t)) + C(x)U(x, t) + D(x) = L C (U(x, t)) + D(x), x, t > U(x, ) = u (x), x, (4.97) where L C = L + C. Then the solution, u, of (4.89), satisfies that u(t) U(t), for all t. Proof. We prove this proposition assuming hypothesis ii., the rest of the cases are analogous, since hypotheses (4.94) and (4.95) imply (4.93), (see proof of Theorem 4.2.9). First of all, we prove that the solution of (4.97) is nonnegative. We know that the solution U can be written with the Variation of Constants Formula as U(t) = e L Ct u + t e L C(t s) D( )ds. (4.98) where L C = L + C = K (h C). Since u, D is nonnegative, and J is nonnegative. If we denote h C = h C, then we can apply Proposition to L C = K h C I, and then we have that e L Ct u t and e L C(t s) D t and s [, t]. Thus, we have that U(t) is nonnegative for all t. Now, we prove that U is a supersolution of (4.89). Since U is nonnegative and f satisfies (4.93), we obtain L(U) + f(, U) L(U) + C( )U + D( ) = U t. Moreover u u = U(), then from Corollary we have u(t) U(t), t. (4.99) Now let W = U be the solution to { W t = L(W) + C( )W D( ) = L C (W) D( ) W() = u. 91

125 Since u then, we have that W(t) = U(t) is nonpositive for all t. Now, we prove that W is a subsolution of (4.89). Since W is nonpositive and f satisfies (4.93), we obtain L(W) + f(, W) L(W) + C( )W D( ) = W t. Moreover, u u = W(), then from Corollary we obtain that Therefore, thanks to (4.99) and (4.1) we have that Thus, the result. u(t) W(t), t. (4.1) U(t) u(t) U(t), t. In the following proposition we give an asymptotic estimate of the norm X of the semigroup of (4.89), S(t)u = u(t, u ), that is given in terms of the norm of the equilibrium associated to the problem (4.97). To obtain this estimate, we assume that the operator L C satisfies that inf σ X ( L C) δ >. (4.11) Then we have that, e L Ct X e δ t for all t. But first we prove a Lemma that will be useful. Lemma Let X be a Banach space, and let S(t) : X X be a continuous semigroup. Assume that u, v X satisfy that S(t)u v in X as t. Then v is an equilibrium point for S(t). Proof. Since v = lim S(t)u. Then applying S(s) for s >, and using the continuity of S(t) t for t >, S(s)v = S(s) lim S(t)u = lim S(s + t)u = v. t t Then v is an equilibrium point for the system. Now, we prove the asymptotic estimate of the solution of (4.97). Proposition Let µ() <, let X = L p (), with 1 p, and h L (). We assume K L(L p (), L ()) is compact (see Proposition 2.1.7), J is nonnegative, f and J satisfy the hypotheses of Proposition 4.3.1, and C L (), D L (). If inf σ X ( L C) δ >, (4.12) then there exists a unique equilibrium solution, Φ, associated to (4.97), such that L(Φ) + C( )Φ + D( ) =, (4.13) Φ L () and Φ. Moreover, if u X, then the solution u of (4.89) satisfies that lim u(t, u ) X Φ X. t 92

126 Proof. First of all, thanks to Proposition 2.4.5, we have that σ X ( L C) is independent of X. Moreover, thanks to hypothesis (4.12), we have that does not belong to the spectrum of L C, then L C = L + C is invertible. Thus, the solution Φ of (4.13) is unique. On the other hand, since Φ satisfies the equation (4.13), D L (), and L C L(L (), L ()), then Φ L (). Now, we want to prove that Φ is nonnegative. We write the solution U with the Variation of Constants Formula U(t) = e L Ct u + Thanks to hypothesis (4.12) and Proposition 3.4.2, we have that then t e L C(t s) D( )ds t. (4.14) e L Ct L(X,X) e δt, (4.15) lim U(t) = e LCs D ds. (4.16) t Thanks to Proposition 3.2.2, we know that e LCt preserves the positivity. From (4.16), since D is nonnegative, then lim U(t). Moreover, thanks to Lemma 4.3.2, we have that lim U(t) t t is an equilibrium, and the problem (4.97) has a unique equilibrium. Then, lim U(t) = Φ, and t Φ is nonnegative. Furthermore, from Proposition 4.3.1, we have that the solution u satisfies that u(t) U(t) = Φ + e L Ct ( u Φ), (4.17) where U(t) is the solution to (4.97). Let us see below that U(t) = Φ + e L Ct ( u Φ) is a solution to (4.97). Since L C = L + C is a linear operator and thanks to (4.13), we have U t (t) = L C (e L Ct ( u Φ)) = L C (U(t) Φ) = L C (U(t)) L C (Φ) = L C (U(t)) + D. For u X, we have that ( u Φ) X, and thanks to (4.15) we obtain u(t) X Since δ >, then from (4.18), we have U(t) X Φ X + e L Ct ( u Φ) X Φ X + e L Ct L(X,X) ( u Φ) X Φ X + e δt ( u Φ) X (4.18) Thus, the result. lim u(t) X Φ X. t Remark The hypotheses on the spectrum of L C, (4.12), can be obtained assuming that J L (, L 1 ()), and h L () satisfies that h C h + δ in, with h (x) = J(x, y)dy L (), and δ >, and J(x, y) = J(y, x). Then, thanks to part ii. of Corollary 2.4.6, we have σ X ( L C) δ >. (4.19) 93

127 Remark Another way to prove that Φ, the equilibrium solution that satisfies (4.13), is nonnegative assuming that J(x, y) = J(y, x) is the following. Thanks to Proposition , we have that L C is selfadjoint in L 2 (). Moreover, we know that Φ L () L 2 (). We multiply (4.13) by Φ := min{φ, } and we integrate in, then we have that D(x) Φ (x)dx = J(x, y)φ(y)dyφ (x)dx+ h(x)φ(x)φ (x)dx C(x)Φ(x)Φ (x)dx = J(x, y)φ(y)dyφ (x)dx+ h(x)(φ ) 2 (x)dx C(x)(Φ ) 2 (x)dx (4.11) In (4.11), we write Φ(y) = Φ + (y) + Φ (y), since Φ + Φ =, thanks to Proposition and the fact that the spectrum of L C satisfies (4.12), we obtain D(x) Φ (x)dx = = J(x, y)(φ + (y) + Φ (y))dyφ (x)dx+ h(x)(φ ) 2 (x)dx C(x)(Φ ) 2 (x)dx J(x, y)φ (y)dyφ (x)dx+ h(x)(φ ) 2 (x)dx C(x)(Φ ) 2 (x)dx = L C (Φ ), Φ L 2 (),L 2 () L C (u), u L 2 (),L 2 () Φ 2 L 2 () inf u L 2 () u 2 L 2 () = inf σ L 2 ()( L C) Φ 2 L 2 () δ Φ 2 L 2 (), (4.111) We have that D and Φ, then thanks to (4.111), we have that D(x)Φ (x)dx δ Φ 2 L 2 () then Φ =. Hence, we have that the solution Φ is nonnegative. 4.4 Extremal equilibria In this section we prove the existence of two ordered extremal equilibria, which give some information about the set that attracts the dynamics of the semigroup S(t) associated to the problem (4.89), S(t)u = u(, t, u ), where u(, t, u ) is the solution of (4.89). A function ϕ = ϕ(x) is said to be an equilibrium solution, or steady-state solution, of (4.89) if it satisfies the following (K hi)(ϕ)(x) + f(x, ϕ(x)) = L(ϕ)(x) + f(x, ϕ(x)) =. (4.112) First of all, we prove the existence of the extremal equilibria for the problem (4.89) with initial data u L (), i.e., we prove that there exists ϕ m and ϕ M in L (), such that the 94

128 solution of (4.89) enter between the extremal equilibria ϕ m and ϕ M for a.e. x, when time goes to infinity. Secondly, we will prove that the same extremal equilibria ϕ m and ϕ M are the bounds of any weak limit in L p () for 1 p < of the solution of (4.89) with initial data u L p (). This is another difference with the nonlinear local problem, with the laplacian, where the asymptotic dynamics of the solution enter between two extremal equilibria, uniformly in space, for bounded sets of initial data (see [44]). This difference is due to the lack of smoothness of the linear group e Lt. We prove now the existence of two ordered extremal equilibria for the problem (4.89) with initial data u L (). Theorem Let (, µ, d) be a metric measure space with µ() <, let X = L p (), with 1 p, and h L (). We assume K L(L p (), L ()) is compact, J is nonnegative, f and J satisfy the hypotheses of Proposition 4.3.1, C L (), D L (), and inf σ X ( L C) δ >. (4.113) Then there exist two ordered extremal equilibria, ϕ m ϕ M, in L () of the problem (4.89), with initial data u L (), such that any other equilibria ψ L () of (4.89) satisfies ϕ m ψ ϕ M. Furthermore, the set {v L () : ϕ m v ϕ M } attracts the dynamics of the solutions S(t)u of the problem (4.89), in the sense that, u L (), there exist u(t) and u(t) in L () such that u(t) u(t, u ) u(t), and in L p () for all 1 p <. lim t u(t) = ϕ m lim t u(t) = ϕ M Proof. From (4.17) we have that the solution of (4.89) satisfies that u(t) Φ + e L Ct ( u Φ) (4.114) Since e L Ct L(L ()) e δt, with δ >, then for every initial data u L () and for all ε >, T (u ) > such that From (4.114) and (4.115) we have that e L Ct ( u Φ) L () < ε, t T (u ). (4.115) Φ ε u(, t, u ) Φ + ε, t T (u ). (4.116) We recall that the solution u of (4.89) is written in terms of the semigroup S(t) as u(, t, u ) = S(t)u. Now, we denote T (u ) = T, to simplify the notation, and we rewrite (4.116) as Φ ε S(t + T )(u ) Φ + ε, t. (4.117) 95

129 In the first part of the proof, we consider the initial data u = Φ + ε, then we have that there exists T = T (Φ + ε) such that Φ ε S(t + T )(Φ + ε) Φ + ε, t. (4.118) Now, thanks to the order preserving properties, Corollary and Corollary , and applying S(T ) to (4.118) with t =, we obtain that Iterating this process, we obtain that Φ ε S(2T )(Φ + ε) S(T )(Φ + ε) Φ + ε. (4.119) Φ ε S(nT )(Φ+ε) S((n 1)T )(Φ+ε) S(T )(Φ+ε) Φ+ε, n N. (4.12) Thus, {S(nT )(Φ + ε)} n N is a monotonically decreasing sequence bounded from below. Then thanks to the Monotone convergence Theorem, the sequence converges in L p (), for 1 p <, to some function ϕ M, i.e. S(nT )(Φ + ε) ϕ M as n in L p (). (4.121) Moreover, since S(nT )(Φ + ε) Φ + ε, for all n N and Φ + ε L (), then ϕ M L (). Now we prove that, in fact, the whole solution S(t)(Φ + ε) converges in L p () to ϕ M as t. From (4.118) we obtain that S(T + t)(φ + ε) Φ + ε, for all t < T. (4.122) Let {t n } n N be a time sequence tending to infinity. We can assume that t n > T. Then If (n + 1)T + t n T, (t n nt ) then S((n + 1)T + t n )(Φ + ε) Φ + ε. (4.123) Applying the semigroup at time t n on (4.123), we have that If t n (n 1)T T, (t n nt ) then S((n + 1)T )(Φ + ε) S(t n )(Φ + ε). (4.124) S(t n (n 1)T )(Φ + ε) Φ + ε (4.125) Applying the semigroup at time (n 1)T on (4.125), we have that S(t n )(Φ + ε) S((n 1)T )(Φ + ε) (4.126) If t n = nt, we have already proved that S(t n )(Φ) converges in L p () to ϕ M as n goes to infinity. Now, let {t n } n N be a general sequence, then taking limits as n goes to infinity in (4.124), we obtain that ϕ M lim inf t n S(t n)(φ + ε) in L p (). 96

130 And taking limits as n goes to infinity in (4.126), we obtain that lim sup S(t n )(Φ + ε) ϕ M t n in L p (). Therefore, lim S(t n)(φ + ε) = ϕ M in L p (). n Since the previous argument is valid for any time sequence {t n } n N we actually have lim S(t)(Φ + ε) = ϕ M in L p (). (4.127) t Now, we prove the result for a general initial data u L (). Thanks to (4.117), for T = T (u ) Φ ε S(t + T )(u ) Φ + ε, t. (4.128) Letting the semigroup act at time t in (4.128), we have Thanks to (4.127) and (4.129), we get that S(T + 2t)u S(t)(Φ + ε) = u(t), t. (4.129) lim u(t) = ϕ M in L p (). (4.13) t Finally, let ψ be another equilibrium. From (4.13), with u = ψ, we get ψ ϕ M. Thus ϕ M is maximal in the set of equilibrium points, i.e., for any equilibrium, ψ, we have ψ ϕ M. The results for ϕ m can be obtained in an analogous way. Corollary Under the hypotheses of the previous Theorem If u L (), and u ϕ M, then lim S(t)(u ) = ϕ M, t in L p (), with 1 p <, i.e., ϕ M is stable from above. In particular this holds for u = Φ. If u L (), and u ϕ m, then and ϕ m is stable from below. lim S(t)(u ) = ϕ m, t in L p (), Proof. As a consequence of Corollary , the associated solutions of (4.89) satisfy S(t)u S(t)ϕ M = ϕ M, t >, (4.131) and from (4.129) and (4.131), there exists T = T (u ), such that Taking limits as t in (4.132), we obtain ϕ M S(T + t)u S(t)(Φ + ε), t >. (4.132) lim S(t)(u ) = ϕ M in L p (). (4.133) t Therefore, ϕ M is stable from above. The proof for ϕ m is analogous. 97

131 Remark If the extremal equilibria ϕ M C b (), then the result of the previous Theorem could be improved because we would obtain the asymptotic dynamics of the solution of (4.89) enter between the extremal equilibria uniformly on compact sets of. Thanks to Dini s Criterium (see [6, p. 194]), we have that S(nT )(Φ + ε) in (4.121), converges uniformly in compact subsets of to ϕ M as n goes to infinity. Thus we have that lim S(t)(Φ + ε) = ϕ M in L t loc (). (4.134) Since there is no regularization for the semigroup S(t) associated to (4.89), we can not assure that ϕ M C b (), as happens for the local reaction diffusion equations. In fact, we give later an example of L () discontinuous equilibria for the problem (4.89), (see example 4.5.7). Now, we want to prove that the previous two ordered extremal equilibria, ϕ m, ϕ M L () in Theorem 4.4.1, are the bounds of any weak limit as t goes to infinity in L p (), of the solution of the problem (4.89) with initial data u L p (). Proposition Let (, µ, d) be a metric measure space with µ() <, let X = L p (), with 1 p <, and h L (). We assume K L(L p (), L ()) is compact, J is nonnegative, f and J satisfy the hypotheses of Proposition 4.3.1, C L (), D L (), and inf σ X ( L C) δ >. (4.135) Then there exist two ordered extremal equilibria, ϕ m ϕ M, in L () of (4.89), with initial data u L p (), with 1 p <. Moreover any other equilibria ψ of (4.89) satisfies ϕ m ψ ϕ M, and the set {v L () : ϕ m v ϕ M } attracts the dynamics of the system, in the sense that for any u L p (), if ũ(, u ) is a weak limit of S(t)u in L p () for 1 p <, when time t goes to infinity, then ϕ m (x) ũ(x, u ) ϕ M (x) for a.e. x. Proof. We consider as initial data, Φ, the equilibrium solution of (4.97). From Corollary we have that the solution to (4.89) with initial data Φ L (), converges in L p () to the maximum equilibrium ϕ M L (), lim S(t)Φ = ϕ M in L p (). (4.136) t On the other hand, thanks to Proposition 4.3.3, we know that given an initial data u L p () S(t)u = u(t, u ) Φ + e L Ct ( u Φ) (4.137) Applying the nonlinear semigroup S(s) to (4.137), and thanks to Proposition 4.1.4, we have that S(s)u(t, u ) = u(t + s, u ) S(s)(Φ + e L Ct ( u Φ)). (4.138) Since the semigroup is continuous in L p () with respect to the initial data, thanks to Proposition 4.3.3, we have the following convergence in L p () lim S(s)(Φ + t elct ( u Φ)) = S(s) lim (Φ + e LCt ( u Φ)) = S(s)Φ. (4.139) t 98

132 Therefore, thanks to (4.138) and (4.139), and since {u(t + s, u )} {t } is bounded in L p (), then let {t n } n N be a sequence that converges to infinity, such that there exists the weak limit in L p () of {u(t n + s, u )} n N when n goes to infinity, denoted by ũ(, u ). In (4.138) we consider t = t n, we multiply (4.138) by ψ L p () and integrate in, then u(x, t n + s, u )ψ(x)dx S(s) ( Φ(x) + e L Ct n ( u (x) Φ(x)) ) ψ(x)dx (4.14) We take limits in (4.14) when n goes to infinity, and thanks to (4.139), we have Then, we have that ũ(x, u )ψ(x)dx S(s)Φ(x)ψ(x)dx. (4.141) ũ(x, u ) S(s)Φ(x), for a.e.x, s >. (4.142) Taking limits now, in (4.142) when s goes to infinity, thanks to (4.136) ũ(x, u ) lim s S(s)Φ(x) = ϕ M (x), for a.e.x. for all u L p (). Thus, the result. The reverse inequality can be proved analogously for the minimal equilibrium ϕ m. The following proposition proves that under the hypotheses of Proposition 4.3.3, and if f(, ), then the maximal equilibria ϕ M, in Theorem is nonnegative. In fact, if J satisfies the hypotheses of Proposition , then any nontrivial nonnegative equilibria, ψ, of the problem (4.89), is strictly positive. Proposition Let (, µ, d) be a metric measure space with µ() <, let X = L p (), with 1 p <, and h L (). We assume K L(L p (), L ()) is compact, J is nonnegative, C L (), D L (), and inf σ X ( L C) δ >. (4.143) If f and J satisfy the hypotheses of Proposition 4.3.1, and f satisfies also that f(, ), then the extremal equilibria of (4.89), ϕ M. Furthermore, if J satisfies that J(x, y) >, x, y such that d(x, y) < R, (4.144) for some R >, and is R-connected, (see Definition ), then any nontrivial nonnegative equilibria ψ of (4.89) is in fact strictly positive. 99

133 Proof. Since f(, ), then is a subsolution of (4.89). Under any of the hypotheses on f in Proposition 4.3.1, thanks to Corollary 4.1.9, Corollary and Corollary , respectively, we know that the subsolutions of the problem (4.89) are below the solution, u(, t, u ), of the problem (4.89), as long as the subsolution exists. Thus, we have that if u, then u(x, t; u ), x, t. (4.145) From Proposition 4.3.3, we know that the solution of (4.13), Φ L (), satisfies that Φ. Thanks to (4.145), since Φ, then the solution associated to the initial datum Φ satisfies that From Corollary u(, t, Φ), t. (4.146) lim u(, t, Φ) = ϕ M, in L p (). (4.147) t Taking limits as t goes to infinity in (4.146), and from (4.147), we have ϕ M. Hence, ϕ M is nonnegative. Moreover, if ψ is a nonnegative equilibria of (4.89) and if J satisfies (4.144), then thanks to Corollary then Thus, the result. ψ = u(, t, ψ) > t >. Under the hypotheses of Proposition 4.4.5, the following proposition states that if f satisfies (4.149) and J satisfies (4.144), then there exists a unique nontrivial nonnegative equilibria. Proposition Let (, µ, d) be a metric measure space with µ() <, let X = L p (), with 1 p <, and h L (). We assume K L(L p (), L ()) is compact, J is nonnegative, C L (), D L (), and inf σ X ( L C) δ >. (4.148) If f and J satisfy the hypotheses of Proposition 4.3.1, and f satisfies also that and f(x, s) s f(, ), is monotone in the variable s, x, (4.149) and J(x, y) = J(y, x) satisfies (4.144). Then there exists a unique nontrivial nonnegative equilibrium, ϕ M, of (4.89), and the equilibrium is strictly positive. 1

134 Proof. From Theorem 4.4.1, let ϕ M L () be the maximal equilibria of (4.89). Now, assume that ψ is another nontrivial nonnegative equilibria, then ψ ϕ M. Thus, ψ L (). Since f(, ), and thanks to Proposition 4.4.5, ψ > and ϕ M >. On the other hand, ψ and ϕ M are equilibria, then they satisfy J(x, y)ϕ M (y)dy h(x)ϕ M (x) + f(x, ϕ M ) = (4.15) J(x, y)ψ(y)dy h(x)ψ(x) + f(x, ψ) = (4.151) We have that ϕ M and ψ belong to L () L p (), for 1 p, then multiplying (4.15) by ψ, and (4.151) by ϕ M, and integrating in, J(x, y)ϕ M (y)dyψ(x)dx h(x)ϕ M (x)ψ(x)dx + f(x, ϕ M )ψ(x)dx =, (4.152) J(x, y)ψ(y)dyϕ M (x)dx h(x)ϕ M (x)ψ(x)dx + f(x, ψ)ϕ M (x)dx =, (4.153) We substract (4.153) from (4.152). Then, we obtain J(x, y)ϕ M (y)dyψ(x)dx J(x, y)ψ(y)dyϕ M (x)dx + f(x, ϕ M ) + ϕ M (x) ϕ f(x, ψ) M(x)ψ(x)dx ψ(x) ϕ M(x)ψ(x)dx =. Relabeling variables in the first term, we have J(y, x)ϕ M (x)dxψ(y)dy f(x, ϕ M ) + ϕ M (x) ϕ M(x)ψ(x)dx Since J(x, y) = J(y, x), we obtain ( f(x, ϕm ) ϕ M (x) J(x, y)ψ(y)dyϕ M (x)dx + f(x, ψ) ψ(x) ϕ M(x)ψ(x)dx =. ) f(x, ψ) ϕ M (x)ψ(x)dx = ψ(x) f(x, s) Moreover, is monotone in the variable s, then, we have that f(x, ϕ M) f(x, ψ) s ϕ M (x) ψ(x) on sets with positive measure. Moreover ϕ M and ψ are strictly positive. Hence, ψ = ϕ M. Thus, the result. Remark Let e Lt be the linear semigroup. We know from Theorem 3.3.4, that e Lt is asymptotically smooth. The nonlinear semigroup associated to (4.89) is denoted by S(t), and given by S(t)u (x) = e Lt u (x) + t e L(t s) f(x, u(x, s)) ds. (4.154) If f : L p () L 1 (), was compact, (which is not), since e L(t s) is a continuous operator, then t el(t s) f(, u(s))ds would be compact, and we would be able to apply [32, Lemma ], to prove that S(t) is asymptotically smooth. But, due to the lack of smoothness of the semigroup e Lt, the semigroup S(t) is not asymptotically smooth in general. 11

135 4.5 Instability results for nonlocal reaction diffusion problem Let (, µ, d) be a metric measure space with µ() <. Let X = L p (), with 1 p. We shall deal with the following nonlocal reaction-diffusion problem, that is the nonlinear problem 4.1 with h = h, and reaction term f, depending only on u, u t (x, t) = J(x, y) (u(y, t) u(x, t)) dy + f(u(x, t)), x, t > (4.155) u(x, ) = u (x), x, where we assume J(x, y) = J(y, x), J, h = J(, y)dy L (), and f : R R, f C 1 (R) is globally Lipschitz. The equation (4.155) can be rewritten as u t = (K h I)u + f(u). It shall be shown that if f is convex or concave then any continuous nonconstant solution of (4.155) is, if it exists, unstable, in some sense to be made precise below. We first introduce a concept of Lyapunov stability with respect to the norm in X. Definition Let u(x, t, u ) be the solution to (4.155) with initial data u X. An equilibrium solution u is Lyapunov stable is for each ε >, there exists δ > such that, if u X and u u X < δ, then u(, t, u ) u X < ε, t >. An equilibrium is unstable if it is not stable. Let us define the concept of instability defined with respect to linearization of the problem (4.155). The linearization of (4.155) around the equilibrium ū is given by ϕ t (x, t) = J(x, y) (ϕ(y, t) ϕ(x, t)) dy + f (u(x))ϕ(x, t), x, t > (4.156) ϕ(x, ) = ϕ (x), x. Since f is globally Lipchitz, then f (ū) L (). Definition The equilibrium u is stable with respect to linearization if for each initial data ϕ X, the solution of (4.156) satisfies sup ϕ(t, ϕ ) X <. t> The equilibrium u is asymptotically stable with respect to linearization if it is stable and if for any initial data ϕ X, the solution of (4.156) satisfies lim ϕ(x, t, ϕ ) = in X. t The equilibrium u is unstable with respect to linearization if there exists an initial data ϕ X, such that the solution of (4.156) satisfies sup ϕ(t, ϕ ) X = +. t> 12

136 If F : X X is C 1, then the stability from linearization implies the stability in the sense of Lyapunov, (see [33, p. 266]). In fact, let A L(X, X), and let F : X X be C 1. We consider the problem u t = Au + F (u) u X. (4.157) Let ū be the equilibrium of (4.157), then We rewrite (4.157) as follows Aū + F (ū) = (4.158) u t = Au + F (u) = A(u ū) + Aū + F (ū) + DF (ū)(u ū) + g(u ū), (4.159) with g(u ū) = F (u) F (ū) DF (ū)(u ū). If we consider v = u ū, then v satisfies v t = Av + DF (ū)v + g(v). (4.16) Since F : X X is C 1, then g(v) X = o( v X ). Let us consider the linearization of the problem (4.157) around the equilibrium ū. ϕ t = Aϕ + DF (ū)ϕ. (4.161) In fact, if u ū X << 1 then g(u ū) X << 1, and the problems (4.16) and (4.161) are almost equal. Therefore, if F : X X is C 1, the stability of the equilibrium of (4.157) can be studied in terms of the stability of the linearized problem (4.161). Hence, we have that the stability from linearization implies the stability in the sense of Lyapunov. On the other hand, we know that the Nemitcky operator associated to f C 1 (R), F : L p () L p () is Lipschitz, but it is not C 1, unless it is linear, (see Appendix B). Hence F : L p () L p () Lipschitz is not differentiable. The following result gives conditions on f under which the stability/instability respect to the linearization (4.156) implies the stability/instability in the sense of Lyapunov, even if F : L p () L p () is not differentiable. Proposition Let (, µ, d) be a metric measure space, with µ() <, X = L p (), with 1 p, and K J L(X, X), h L (), and let a < c < d < b. We assume J nonnegative, f C 2 (R), nonlinear and globally Lipschitz, and ū L () is an equilibrium solution of (4.155) with values in [c, d]. i. If f > in [a, b], and the equilibrium ū is unstable with respect to the linearization then ū is unstable in the sense of Lyapunov in X. ii. Let X = L p (), with 1 p, let K L(L p (), L ()) be compact and h L (). If f ( < in [a, b], σ X K (h f (ū))i ) δ <, and the equilibrium ū is asymptotically stable respect to the linearization then ū is stable from above, in the sense that, if an initial datum u takes values in [a, b] and satisfies that u ū, then the solution of (4.155) with initial datum u converges to ū in X when time goes to infinity. 13

137 Proof. Let z be the solution of the nonlinear problem z t (x, t) = J(x, y)(z(y, t) z(x, t))dy + f(z(x, t)), x, t >. z(x, ) = u (x), x. (4.162) with u L (), and let ū be the equilibrium solution of (4.162), then J(x, y)(ū(x) ū(y))dy + f(ū(x)) =. (4.163) Let us consider the linearization of (4.162) around ū, ϕ t (x, t) = J(x, y) (ϕ(y, t) ϕ(x, t)) dy + f (ū(x))ϕ(x, t) = L(ϕ)(x, t), x, t >, ϕ(x, ) = ϕ (x), x, then we consider From (4.165) and (4.162), v satisfies v t (x, t) = J(x, y) (v(y, t) v(x, t)) dy + v(x, ) = u (x) ū(x). i. Since f > in [a, b], we have that f satisfies that (4.164) z(x, t) = ū(x) + v(x, t). (4.165) J(x, y) (ū(y) ū(x)) dy + f ( ū(x) + v(x, t) ). (4.166) f(ū + v) f(ū) + f (ū)v, (4.167) for v small enough such that ū + v takes values in [a, b]. Applying inequality (4.167) to (4.166), we obtain that v t (x, t) J(x, y)(v(y, t) v(x, t)) dy + J(x, y)(ū(y) ū(x)) dy + f(ū(x)) + f (ū(x))v(x, t), (4.168) for all t such that ū + v(t) takes values in [a, b]. Since ū is an equilibrium, it satisfies equality (4.163), then v t (x, t) J(x, y) (v(y, t) v(x, t)) dy + f (ū(x))v(x, t). (4.169) for all t such that ū + v(t) takes values in [a, b]. If v() ϕ, then v is a supersolution of (4.164), and from Proposition v(x, t) ϕ(x, t), x, for t > such that ū + v(t) takes values in [a, b]. (4.17) Since ū is unstable with respect to the linearization, then we prove below that there exists ϕ L (), with ϕ > such that sup ϕ(t, ϕ ) X = +. (4.171) t> 14

138 Let us prove that there exists ϕ > that satifies (4.171). First, we argue by contradiction in X = L (), then for all ϕ L () with ϕ, we have that sup t> ϕ(t, ϕ ) L () <, and since (4.164) is a linear problem, we have that for all ϕ, sup t> ϕ(t, ϕ ) L () <. Hence, for any initial data ϕ = ϕ + ϕ, it happens that ϕ(t, ϕ ) = ϕ(t, ϕ + ) ϕ(t, ϕ ), and ϕ(t, ϕ ) L () <. Arguing by density we obtain that ϕ(t, ϕ ) X <. Thus, we arrive to contradiction with the fact that ū is unstable with respect to the linearization. Thanks to Proposition we have that if ϕ, then ϕ(x, t, ϕ ), for all x and t >. Since ϕ is nonnegative and from (4.17) and (4.171) we have that C v(t) L () v(t) X ϕ(t, ϕ ) X, (4.172) for all t such that ū + v(t) takes values in [a, b]. On the other hand from (4.171), for all δ >, there exists µ > such that µϕ < δ, and there exists t such that ϕ(t, µϕ ) X max{ a, b }. (4.173) Hence, thanks to (4.172) and (4.173), for all δ >, if we choose v() = µϕ as above, then v() X = u ū X < δ, and there exists t > such that z(t ) ū X = v(t ) X ϕ(t, ϕ ) X max{ a, b }. Hence, the equilibrium ū is Lyapunov unstable. ii. Since f < in [a, b], we have that f satisfies that f(ū + v) f(ū) + f (ū)v, (4.174) for v small enough such that ū+v takes values in [a, b]. Applying inequality (4.174) to (4.166), we obtain that v t (x, t) J(x, y)(v(y, t) v(x, t)) dy + J(x, y)(ū(y) ū(x)) dy + f(ū(x)) + f (ū(x))v(x, t), (4.175) for all t such that ū + v(t) takes values in [a, b]. Since ū is an equilibrium, it satisfies equality (4.163), then from (4.175) we have v t (x, t) J(x, y) (v(y, t) v(x, t)) dy + f (ū(x))v(x, t), (4.176) for all t such that ū + v(t) takes values in [a, b]. Thus, if v() ϕ, then v is a subsolution of (4.164), and from Proposition v(x, t) ϕ(x, t), for t > such that ū + v(t) takes values in [a, b]. (4.177) Since we want to prove the stability from above, we consider an initial datum u ū, then thanks to Proposition , we know that z(x, t, u ) z(x, t, ū) = ū(x) for all x for all t >, then v(x, t, u ū) = z(x, t, u ) ū(x) for all x for all t >. 15

139 Let us prove that under the hypotheses in the statement, ū + v(t) takes values in [a, b] for all t. If ϕ, thanks to Proposition 3.2.2, we have that ϕ(t, ϕ ), for all t. Moreover, from (4.177) and since v(t, u ū) for all t, we have ū ū + v(t) ū + ϕ(t), for t > such that ū + v(t) takes values in [a, b] (4.178) Moreover, from (4.178), we have that a ū + v(t) b, for all t, if ū + ϕ(t) b, for all t, i.e., if ϕ(t) L () b inf ū(x) = b d, for all t. Thanks to Proposition 2.4.5, σ X (K (h f (ū))i) is independent of X. Moreover, since σ X (K (h f (ū))i) δ < and thanks to Proposition 3.4.2, then ϕ(t, ϕ ) L () C e δt ϕ L () C ϕ L (), for all t. Hence if we choose an initial datum ϕ L (), such that C ϕ L () b d, then ϕ(t, ϕ ) L () b d for all t. Thus, a ū + v(t) b, for all t, and thanks to (4.177), we obtain that v(x, t) ϕ(x, t), for all t. (4.179) Furthermore, since ū is asymptotically stable with respect to the linearization, then for any initial data ϕ L () lim ϕ(, t, ϕ ) L t () =. (4.18) If we choose an initial data small enough such that v() ϕ, with ϕ satisfying C ϕ L () b d, then from (4.179), (4.18), and since v(t), we have lim v(, t, v()) L t () = (4.181) Furthermore, since z(x, t) = ū(x) + v(x, t) then z converges to ū in L () when t goes to. Since we have the convergence in L (), and µ() <, we have also the convergence in X. Thus, the result. In the following result we give a criterium to prove that an equilibrium ū is unstable with respect to the linearization. Theorem Let (, µ, d) be a metric measure space, with µ() <. For 1 p 2, let X = L p (), with p p, and we assume K L(L p (), L ()) is compact, J nonnegative, f C 2 (R) nonlinear and globally Lipschitz, and u L () is an equilibrium of (4.155). For ϕ L 2 (), we define I(ϕ) = 1 J(x, y)(ϕ(y) ϕ(x)) 2 dy dx + f (u(x))ϕ 2 (x)dx. 2 If there exists ϕ L 2 () such that I(ϕ) >, then u is unstable with respect to linearization in X. Proof. Multiplying (4.156) by ϕ and integrating in, we obtain ϕ t (x, t)ϕ(x, t)dx = J(x, y) (ϕ(y, t) ϕ(x, t)) dyϕ(x, t)dx + 16 f (u(x))ϕ 2 (x, t)dx.

140 Thanks to Proposition (Green s formula), t Thus, we denote ϕ 2 (x, t) dx = 1 J(x, y)(ϕ(y, t) ϕ(x, t)) 2 dy dx + f (u(x))ϕ 2 (x, t)dx 2 2 I(ϕ) = 1 2 J(x, y)(ϕ(y) ϕ(x)) 2 dy dx + f (u(x))ϕ 2 (x)dx. Now we assume that there exists ϕ such that I(ϕ) >. We define λ = sup I(ϕ) I(ϕ) >. ϕ L 2, ϕ =1 Thanks to Proposition , λ > belongs to the spectrum of L = K (h f (ū))i in L 2 (). Moreover, thanks to the hypotheses, and Proposition 2.4.5, λ σ X ( L). Now we prove that ū is unstable with respect to the linearization. We argue by contradiction, we assume that ū is stable with respect to the linearization, then for any ϕ X, ϕ(t, ϕ ) X <, for all t, i.e. for any ϕ X, e e Lt ϕ X M(ϕ ), for all t, then applying Banach-Steinhaus Theorem to the family of operators {e e Lt } t, we have that there exists M such that e e Lt L(X,X) M for all t. Hence, for all ε >, e (e L ε)t L(X,X) e εt M (4.182) Furthermore for all λ >, the resolvent can be written as follows, (see [24, p. 614]), Therefore, from (4.182) and (4.183) we have that (λi + εi L) 1 L(X,X) ( 1 (λ + ε)i L) = e (e L ε)t e λt dt. (4.183) e (e L ε)t e λt L(X,X) dt M e ( λ ε)t dt = M 1 λ + ε. Then (λi + εi L) 1 L(X, X) for all λ >. Then {λ R + : λ > ε} ρ X ( L) for all ε >. Hence, R + ρ X ( L), and we arrive to contradiction with the fact that λ σ X ( L) and λ >. Therefore, ū is unstable with respect to the linearization. The zeros of f are the constant equilibriums of (4.155), and thanks to the criterium of the previous Theorem 4.5.4, we have that a constant equilibrium ū is unstable if there exists ϕ such that I(ϕ) = 1 J(x, y)(ϕ(y) ϕ(x)) 2 dy dx + f (u(x))ϕ 2 (x)dx >. 2 Thus, if we take ϕ constant, we will have that I(ϕ) = f (u)ϕ 2 dx. 17

141 Therefore, a constant equilibrium ū is unstable with respect to the linearization if f (u) >. In the Theorem below, we find conditions guaranteeing that for a nonconstant equilibrium u, there exists ϕ such that I(ϕ) >. The instability results depend on the function f. First we make the observation that if u is a nonconstant equilibrium such that f (u(x))dx > then u is unstable with respect to the linearization (4.156). This follows from the fact that I(ϕ) > for ϕ 1. The following result states that if the function f is strictly convex or strictly concave, then any continuous and bounded nonconstant solution is unstable with respect to the linearization. Theorem Let (, µ, d) be a metric measure space, with µ() <. For 1 p 2, let X = L p (), with p p, and let a < c < d < b. We assume K L(L p (), L ()) is compact, J nonnegative, f C 2 (R) nonlinear and globally Lipschitz. Let ū C b () be a nonconstant equilibrium solution of (4.155) with values in [c, d]. If either f > on [a, b] or f < on [a, b], then u is unstable with respect to the linearization in X. Proof. Consider first the case f >. Let c = inf ū(x), then we establish instability by x showing that I(ū c) >, and applying Theorem Now I(ū c) = 1 J(x, y)(ū(y) ū(x)) 2 dy dx + f (ū(x))(ū(x) c) 2 dx. (4.184) 2 Since ū is an equilibrium solution of (4.155), J(x, y)ū(y)dy J(x, y)dy ū(x) + f(ū(x)) =. (4.185) Integrating (4.185) in = f(ū(x))dx = J(x, y)ū(y)dy J(x, y)ū(y)dy dx J(x, y)dy ū(x)dx + f(ū(x))dx J(x, y)ū(x)dy dx. Since J(x, y) = J(y, x), and relabeling variables, we get f(ū(x))dx = J(x, y)ū(y)dy dx J(y, x)ū(y)dy dx =. (4.186) Now, multiplying (4.185) by ū, integrating in and thanks to Proposition 2.3.1, (Green s formula), we obtain f(ū(x))ū(x) dx = 1 J(x, y)(ū(y) ū(x)) 2 dy dx. (4.187) 2 18

142 From (4.184), (4.186) and (4.187), I(ū c) = (ū(x) c) [ f(ū(x)) f (ū(x))(ū(x) c) ] dx. (4.188) Now, we prove that f(c). Since ū C b () is an equilibrium solution, ū satisfies the equality (4.185), and considering that x {x : ū(x) = c}, then f(c) = J( x, y)(c ū(y))dy. From the condition on f we have that if ū(x) c, then f(c) > f(ū(x)) + f (ū(x))(c ū(x)). Since f(c), then > f(ū(x)) f (ū(x))(ū(x) c). Moreover, since ū is nonconstant, if ū(x) > c = inf ū(x), then I(ū c), given by (4.188), satisfies that I(ū c) >. x The proof of the case when f < follows in a similar argument except now we take c = max ū(x) and note that when f <, we will have f(c). x Corollary Under the hypotheses of Theorem Let ū C b () be a nonconstant equilibrium solution of (4.155) with values in [c, d]. If f satisfies that f > on [a, b] [c, d], then ū is unstable in the sense of Lyapunov. Proof. From Theorem 4.5.5, we know that if f >, then the nonconstant equilibrium ū is unstable with respect to linearization. And thanks to Proposition 4.5.3, if f >, and ū is unstable with respect to linearization, then it is unstable in the sense of Lyapunov. Thus, the result. Remark (Example of non-isolated and discontinuous equilibria) We construct a particular example of the problem (4.155), in which we give an explicit expression for nonisolated and discontinuous equilibria. This is different from the local problem, since for the local reaction-diffusion problem the equilibria are continuous, thanks to the regularization of the semigroup associated to. If we choose J(x, y) = 1, for all x, y, and f(u) = λu(u 2 1), then the equilibria of (4.155) satisfy J(x, y)(u(y) u(x))dy + f(u(x)) =, then u(y)dy = µ()u λu(u 2 1). (4.189) The left-hand side of (4.189) is We denote the right-hand side of (4.189) by u(y)dy = A, with A R g(u) = µ()u λu(u 2 1). 19

143 Hence, given A, we take the solutions u of g(u) = A. In figure 4.1, we can see a particular example, in which there are three different roots, that satisfy g(u) = A, and we denote them by u 1, u 2, u 3. If the divide the set in three arbitrary subsets 1, 2, 3, then we can construct the equilibria u(x) = u 1 χ 1 (x) + u 2 χ 2 (x) + u 3 χ 3 (x). This family of equilibria is not isolated, because we can build a new partition of, denoted by 1, 2, 3, and we consider the equilibrium ũ(x) = u 1 χ e1 (x) + u 2 χ e2 (x) + u 3 χ e3 (x), such that ũ is as close as we want, in L p (), to the equilibrium u. Figure 4.1: Roots of A = g(u) 11