II Mini-Workshop in Partial Differential Equations

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1 Campus de São José do Rio Preto Programa de Pós-Graduação em Matemática (PPGMAT-UNESP) II Mini-Workshop in Partial Differential Equations August Program: 10:00-10:30 -Fluidos micropolares com dados iniciais em espaços de Besov-Morrey Juliana Precioso (IBILCE/UNESP) 10:35-11:05 -Rate of continuity of attractors for a parabolic problem discretized via finite element Rodiak Figueroa López (IBILCE/UNESP) 11:10-11:40 -Almost periodicity for a class of neutral functional differential equations Andréa Prokopczyk Arita (IBILCE/UNESP) :00-14:30 -Viscous Cahn-Hilliard equation in R N Tomasz Dlotko (University of Silesia, Katowice, Poland) 14:35-15:05 -On the concept of attractors for non-autonomous dynamical systems José Antonio Langa Rosado (Universidade de Sevilla, Espanha) 15:10-16:40 -Morse-Smale non-autonomous dynamical systems Matheus Cheque Bortolan (ICMC-USP, São Carlos, Brasil) Local: Sala de Seminários do Departamento de Matemática Organizers: Germán Lozada Cruz (IBILCE/UNESP) Juliana Precioso (IBILCE/UNESP) Andréa Prokopczyk Arita (IBILCE/UNESP)

2 ABSTRACTS Fluidos micropolares com dados iniciais em espaços de Besov-Morrey Juliana Precioso (IBILCE, UNESP) Nesta palestra, mostraremos um resultado de existência local no tempo para o sistema de fluidos micropolares tridimensionais no âmbito dos espaços de Besov-Morrey. A classe de dados iniciais é maximal, isto é, é maior do que as anteriores e contém funções fortemente singulares e medidas. Almost periodicity for a class of neutral functional differential equations Andréa Prokopczyk Arita (IBILCE, UNESP) In this work we study the existence of asymptotically almost periodic solutions for a class of second order abstract neutral differential equations of the form d 2 dt 2 [x(t) + g(t, x t)] = Ax(t) + f(t, x t ), t [0, ), (0.1) x 0 = ϕ B, (0.2) x (0) = ξ X, (0.3) where A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on a Banach space (X, ), (C(t)) t R, the history x t : (, 0] X, x t (θ) = x(t + θ), belongs to an abstract phase space B defined axiomatically and f, g are suitable functions. References [1] Diagana, T., Henríquez, H. and Hernández, E. Asymptotically almost periodic solutions to some classes of second order functional differential equations, Dif. Integral Equations 21 (2008), no. 5-6, [2] Henríquez, H. R. and Vásquez, C. Almost periodic solutionsof abstract retarded functional differential equations with unbounded delay, Acta Appl. Math. 57 (1999), [3] Hernández, E. and Henríquez, H. R. Existence results for second order differential equations with nonlocal conditions in Banach spaces, Funkcialaj Ekvacioj 52 (2009), [4] Hernández, E., Henríquez, H. R. and McKibben, M. A. Existence of solutions for second order partial neutral functional differential equations, Integral Equations Operator Theory 62 (2008), no. 2, [5] Zaidman, S. Almost periodic functions in abstract spaces Res. Notes in Math Pitman (Advanced Publishing Program), Boston, MA,

3 Rate of continuity of attractors for a parabolic problem discretized via finite element Rodiak Figueroa López (IBILCE, UNESP) In this work we consider the parabolic problem u t = Lu + f(u), t > 0, x Ω u(t, x) = 0, t > 0, x Ω u(0, x) = u 0 (x), x Ω, (0.4) where Lu = n i,j=1 ( aij (x) u x i x j ) + n j=1 b j (x) u x j + (c(x) + λ)u is a second order operator with uniformly strongly elliptic condition, u 0 H0 1 (Ω), Ω Rn is a bounded domain with smooth boundary, n 1, a ij, b j, c : Ω R are smooth functions, λ R and f C 2 (R). Under certain growth and dissipativity conditions we have the existence of an global attractor A for (0.4) in a Banach space L 2 (Ω). Our interest in this work is to study the robustness of attractor A under discretization by finite element method. In particular to study the continuity of the family of attractors {A h } h (0,1] when the global step size goes to zero. For this we use the concept of P convergence given in [4]. References [1] Arrieta, José M.; Carvalho, Alexandre N.; Rodríguez-Bernal, Aníbal Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds. Comm. Partial Differential Equations 25 (2000), n. 1-2, p [2] Arrieta, J.M.; Carvalho, A.N., Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain. Journal Differential Equations 199 (2004), n. 1, p [3] Arrieta, J.M.; Carvalho, A. N.; Lozada-Cruz, G., Dynamics in dumbbell domains. I. Continuity of the set of equilibria. Journal Differential Equations 231 (2006), n. 2, [4] Vainikko G. Funktionalanalysis der Diskretisierungsmethoden, Teubner-Texte zur Mathematik, Verlagsgesellschaft, Leipzig,

4 Viscous Cahn-Hilliard equation in R N Tomasz Dlotko (University of Silesia, Katowice, Poland) Abstract Solvability of Cauchy s problem in R N for an extended viscous Cahn-Hilliard equation will be discussed. The problem is considered first in a standard Sobolev space H 1 (R N ), next a notion of the H-solution is introduced which is well adapted to the structure of the viscous Cahn-Hilliard equation. Several properties of an unbounded operator ( ) 1 in R N needed in our considerations will be also reported. The Cauchy problem for an extended viscous Cahn-Hilliard equation has the form: (1) { (1 ν)ut = ( u + f(x, u) νu t ), t > 0, x R N, u(0, x) = u 0 (x), where ν [0, 1) and the nonlinear term f fulfills the required regularity and growth assumptions. References [1] L.A. Caffarelli, N.E. Muler, An L bound for solutions of the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 133 (1995), [2] A.N. Carvalho, T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl. 344 (2008), [3] L. Cherfils, A. Miranville, S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), [4] T. Dlotko, C. Sun, Dynamics of the modified viscous Cahn-Hilliard equation in R N, Topol. Methods Nonlinear Anal. 35 (2010), [5] T. Dlotko, M.B. Kania, Chunyou Sun, Analysis of the viscous Cahn-Hilliard equation in R N, J. Differential Equations 252, 2012,

5 On the concept of attractors for non-autonomous dynamical systems José A. Langa Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla Abstract Dynamical systems theory allows the modelization and study of multiple phenomena of Natural and Social Sciences. When the models are characterized by partial differential equations, the theory of global attractors for infinitedimensional dynamical systems has been used during the last fifty years as the central object to study some of these phenomena. However, in the last two decades an intensive research has been done when time-dependent (or even random) terms are needed for the mathematical analysis of some real phenomena, described by the so-called non-autonomous dynamical systems. The dynamical properties of these extended dynamical systems is much richer, so that new concepts and tools have to be introduced and developed. In this talk we will try to describe the main topics related to this new area of research ([1], [2]), paying special attention to the theoretical aspects of the theory. Referencias [1] Carvalho, A.N, Langa, J.A., Robinson, J.C., Attractors for infinite-dimensional nonautonmous dynamical systems, Applied Mathematical Series, Springer, New York [2] Kloeden, P.E., Rasmussen, M., Nonautonomous dynamical systems. Mathematical surveys and monographs. AMS, Providence, RI

6 Morse-Smale Non-Autonomous Dynamical Systems Matheus Cheque Bortolan (ICMC-USP, São Carlos, Brasil) In this lecture we define non-autonomous Morse-Smale dynamical systems and prove the phase diagram commutativity between attractors of a Morse-Smale semigroup and its non-autonomous perturbation. 6

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