# EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp ISSN: UL: or ftp ejde.math.txstate.edu (login: ftp) EXISTENCE AND NON-EXISTENCE ESULTS FO A NONLINEA HEAT EQUATION CANAN CELIK Abstract. In this study, we consider the nonlinear heat equation u t(x, t) = u(x, t) + u(x, t) p in Ω (, T ), Bu(x, t) = on Ω (, T ), u(x, ) = u (x) in Ω, with Dirichlet and mixed boundary conditions, where Ω n is a smooth bounded domain and p = + /n is the critical exponent. For an initial condition u L, we prove the non-existence of local solution in L for the mixed boundary condition. Our proof is based on comparison principle for Dirichlet and mixed boundary value problems. We also establish the global existence in L +ɛ to the Dirichlet problem, for any fixed ɛ > with u +ɛ sufficiently small.. Introduction In this paper we study the existence and non-existence results for the initial and boundary value problems of the form u t (x, t) = u(x, t) + u(x, t) p u(x, t) in Ω (, T ), Bu(x, t) = on Ω (, T ), u(x, ) = u (x) in Ω, (.) where Ω n is a smooth bounded domain, B denotes the corresponding boundary condition and p is the critical exponent, which will be specified later. Throughout this paper we use the following definition for the solution. Definition.. A function u = u(x, t) is a mild solution of initial and boundary value problem (.) in Ω [, T ], T being a positive number, if and only if u C([, T ], L q (Ω)) satisfies the integral equation t u(x, t) = K(x, y, t)u (y)dy + K(x, y, t s) u(y, s) p u(y, s) dy ds, (.) Ω Ω Mathematics Subject Classification. 35K55, 35K5, 35K57, 35B33. Key words and phrases. Nonlinear heat equation; mixed boundary condition; global existence; critical exponent. c 7 Texas State University - San Marcos. Published February 8, 7. 5

2 5 C. CELIK EJDE/CONF/5 where u L q (Ω) and K denotes the heat kernel for the linear heat equation with Dirichlet boundary condition. It is well known that if the initial condition u L (Ω), there exists a unique solution of (.) on [, T max ). However the question of local existence and uniqueness was interesting when u / L (Ω); i.e., u L q (Ω) where q <. This type of initial and boundary value problem was studied by many authors. First Fujita [, 5] studied this problem for classical solutions and established the following results for the Cauchy problem u t (x, t) = u(x, t) + u p (x, t) in n (, T ), u(x, t) = u (x) on n, (.3) where u (x). (i) If n(p )/ <, no non-negative global solution exists for any non-trivial initial data u L. That is, every positive solution to this initial value problem blows up in a finite time. (ii) If n(p )/ >, global solution do exist for small initial data. To be precise, for any k >, δ can be chosen such that problem (.3) has a global solution whenever u (x) δe k x. Later Hayakawa [8] showed that the critical case for the Cauchy problem (.3), that is n(p )/ =, also belongs to blow up case for n =,. After Hayakawa the same result was proved by Kobayashi, Sirao and Tanaka [9] for general n. Also Weissler [] obtained that for n(p )/ with p <, non-negative L p solutions to the Cauchy problem blow up in L p norm in finite time. However in the case n(p )/ >, he obtained sufficient conditions for the initial data u which guarantee the existence of global solution. The initial and boundary value problem was considered by Weissler [3, ] who obtained some important local existence and uniqueness results in C([, T ], L q (Ω)). In [3], he considered the case q > n(p )/ and q > p and proved the local existence and uniqueness in C([, T ], L q (Ω)) for which the same argument works for q > n(p )/ and q = p. Later local existence in C([, T ], L q (Ω)) with q n(p )/ and q > or q > n(p )/ and q was proved in []. In the same study he also proved a uniqueness result in a smaller class which was improved in []. If we notice, while the applications of the abstract results in [3] and [] are given on a bounded domain, they work equally well on all of n. The first non-uniqueness result for this problem was given by Haraux-Weissler in [7]. Further non-uniqueness results which were motivated by [7] came in [] and []. A unique solution to the initial and boundary value problem (.) in L p (, T ; L p ) was constructed by Giga [6]. In his work, the main relation between p and p was p = n ( r p ), p > r, provided that the initial data u L r with r = n(p )/ >. As we stated above, it is well known that if u L (Ω), there exists a unique solution defined on a maximal interval [,T max ) and Brezis and Cazenave [] considered the case if u / L (Ω); i.e., u L q (Ω) for some q <. And they studied two cases. (i) If q > n(p )/ (resp. q = n(p )/) and q (resp. q > ) with n, they obtained existence and uniqueness of a local solution in

3 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 53 C([, T ], L q (Ω)) which is a classical solution of the initial boundary value problem on Ω (, T ). (ii) If q < n(p )/, they showed that there exists no local solution in any reasonable sense for some initial data u L q (Ω). emark.. The quantity q = n(p )/ plays a critical role for this initial and boundary value problem. To see that q = n(p )/ is the critical exponent, we observe the following dilation argument. If v is the solution of v t = v xx + v p, v(x, ) = g(x), then u(x, t) = k s v(kx, k t) is the solution of u t = u xx + u p, u(x, ) = k s g(kx). By using the equation u t = u xx + u p with u(x, t) = k s v(kx, k t), we get k s+ v t = k s+ v xx + k sp v p which gives the condition s = /(p ). Also note that, u() q q = u q (x, )dx = k sq n v q q (y, )dy = v() q when s = n/q. So, combining these two conditions for s, we have q = n(p )/, which is the critical exponent. The question of existence and uniqueness of a local solution for the doubly critical case which is q = n(p )/ and q =, which was proposed by Brezis and Cazenave [], was wide open even for n =,(p = 3). In [3], we proved that for u = j= j e je (ej )x, Dirichlet problem has no non-negative local solution in L (, ). Moreover we generalized the non-existence result of local solution for n-dimensional case with the critical exponent p = + n. More general nonlinearity was also considered for Dirichlet boundary value problem. This paper is organized as follows: In Section, we summarize the non-existence results of local solution for both cauchy and the Dirichlet problems for doubly critical case q = n(p )/ and q =, first for one dimensional case and then n-dimensional case. In Section 3, we consider the same problem with mixed boundary conditions; that means having u and the normal derivative of u in the boundary condition. We prove the non-existence of local solution for the mixed boundary conditions by using the comparison argument for the kernels with Dirichlet and mixed boundary conditions. In Section, we give the sufficient conditions for q and the initial data u to guarantee the existence of the global solution to the Dirichlet problem. In particular, we prove the global existence in L +ɛ with u +ɛ sufficiently small and ɛ >, for the critical exponent p = 3.. Non-existence of a Local Solution In the following two sections, we summarize the non-existence of local solution for both cauchy and Dirichlet problems.

4 5 C. CELIK EJDE/CONF/5.. Non-existence of Local Solutions for the Cauchy Problem. To motivate the proof of the non-existence of local L solution for the Dirichlet problem, we first prove the non-existence of local L solution for the Cauchy problem for the one dimensional case, u t (x, t) = u xx (x, t) + u 3 (x, t) in (, T ), u(x, t) = u (x) on, (.) for some u L (). First we choose a particular initial data u (x) = ke k x L () and observe the following result. After this observation, we prove in Theorem.. that the Cauchy problem has no local L solution for initial data of the form u = k c kke k x where c k will be determined later. We denote the solution of the Cauchy problem (.) by u c. For the Cauchy problem (.) with the initial data u (x) = ke k x L (), using that e (x y) t e ay dy = e ax +at, πt + at the linear solution is u L k c (x, t) = + k t e k x +k t. So the solution to the Cauchy problem (.) is t u c (x, t) = u L c (x, t) + K c (x, y, t s)(u c (y, s)) 3 dy ds, where the Cauchy kernel is Note that K c (x, y, t) = (u L c (y, s)) 3 = x y e t. πt k 3 ( + k s) 3k y e +k s 3/ and K c(x, y, t s)dx =. Since u L c (x, t) > and u c (x, t) > u L c (x, t), we have t u c (x, t) > K c (x, y, t s)(u L c (y, s)) 3 dy ds. Then for any δ >, u c L ( (,δ)) = > = = = δ δ t δ t δ t π 3 u c (x, t) dx dt δ t K c (x, y, t s)(u L c (y, s)) 3 dy ds dx dt (u L c (y, s)) 3 k 3 e 3k y +k s ( + k s) K c (x, y, t s)dx dy ds dt 3/ dy ds dt k ( + k ds dt = s) π δ 3 ln( + k t)dt.

5 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 55 We can use the above estimate to establish the following theorem for the Cauchy problem. Theorem.. For some initial data u L (), u, the Cauchy problem (.) has no non-negative local mild solution in C([, δ], L ()) for all δ >. Proof. Let u = k c kke k x where c k will be determined later. The linear solution to the Cauchy problem (.) is u L c (x, t) = k c k k + k t e k x +k t. Note that (u L c (y, s)) 3 c 3 k k 3 3k y e ( + k +k s s) 3/. k Since the solution of the problem satisfies t u c (x, t) > K c (x, y, t s)(u L c (y, s)) 3 dy ds, we have that for any δ >, u c L ( (,δ)) = > = δ δ u c (x, t) dx dt t δ t δ t = k = k C k k δ t C K c (x, y, t s)(u L c (y, s)) 3 dy ds dx dt (u L c (y, s)) 3 c 3 k k3 ( + k s) c 3 k k ( + k ds dt s) δ C c3 k ln( + k t)dt δ δ/ c 3 k ln( + k t)dt K c (x, y, t s)dx dy ds dt 3k y e 3/ +k s dy ds dt C k δc 3 k ln( + k δ). Now we choose c k s such that (i) k c3 k δ ln( + k δ) = ; and (ii) k c k < so that u L. We will find a sequence {k j } j= such that c k only when k = k j. For k big enough, we have c 3 kδ ln( + k δ) c 3 k ln k. So if we set ck 3 ln k = (ln k) / for some k = k j as j ; that is, if we choose c k =, for k = k (ln k) / j, j =,,, condition (i) will be satisfied for all δ >.

6 56 C. CELIK EJDE/CONF/5 For (ii), we set = (ln k) / j Then for u (x) = which implies that k = e j j= j ej e (ej )x, the Cauchy problem (.) has no local solution in L. = k j, j =,,... Non-existence of Local Solutions for the Dirichlet Problem. In this section we will prove that there is no local L solution to the one dimensional Dirichlet problem u t (x, t) = u xx (x, t) + u 3 (x, t) in (, ) (, T ) u(±, t) =, u(x, ) = u (x) in (, ), (.) for some initial data u L. First by using the lower solution argument with a cutoff function we prove some estimates for a family of particular initial data and then we will construct the initial data u L (, ) for which there is no local L solution to the Dirichlet problem (.). Throughout this section we denote the solution of the Dirichlet problem (.) by u d. Let u (x) = ke k x, then by Fourier series, the linear solution to the problem (.) is where K d (x, y, t) = u L d (x, t) = j= K d (x, y, t)ke k y dy e j π t sin jπ( + y is the Dirichlet kernel. Then the solution of (.) is u d (x, t) = u L d (x, t) + t + x ) sin jπ( ) K d (x, y, t s)u 3 d(y, s) dy ds. The difficulty for the Dirichlet problem is that we do not have a good explicit formula for the solution in a bounded domain as we have for the Cauchy problem. In the proof for the Cauchy problem, K c(x, y, t s)dx = is used to simplify the argument. For the Dirichlet problem we do not have that but we know that Ω K d(x, y, t)dx decays exponentially for t >. The idea behind the proof of the non-existence of local solution for the Dirichlet problem is to use the non-existence of local solution for the Cauchy problem (which was proved in the previous section). To do this, we first construct a cutoff function and estimate the Dirichlet kernel K d in terms of the Cauchy kernel K c. Then using this estimate we find a lower bound for the linear solution of the Dirichlet problem u L d in terms of the linear solution of the Cauchy problem ul c. Combining the estimates for K d and u L d, we can estimate u d L and obtain the non-existence result of local solution for the Dirichlet problem. In the first lemma below, we estimate the Dirichlet kernel K d in terms of the Cauchy kernel K c by constructing a cutoff function in a concentrated domain.

7 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 57 Lemma.. For x y and x, K d (x, y, t) e t ( + (x y) + )K c (x, y, t). Proof. The crux of the proof of this lemma is to construct a cutoff function g so that the function w(x, y, t) = gk c (x, y, t) K d (x, y, t) will satisfy the following conditions: (i) L(w) where L(w) = w t w xx, (ii) w on the boundary x y = with x and at t =. Since L(K d ) =, we want w to satisfy L(w) = L(gK c K d ) = L(gK c ) = L(g)K c K c. g. x y Note that K c (x, y, t) = e t πt K c = implies (x y) t e x y t = πt (x y) K c. t So g must satisfy (x y) L(w) = L(g)K c + K c g ; t that is, (x y) g t g xx + g x t in the one-dimensional case. To satisfy this inequality and to have zero on the boundary (i.e., when y = x ± ), we choose the cutoff function as g(x, y, t) = e αt( + (x y) ) + where α will be determined later. So we want (x y) g t g xx + g x t ( = α + (x y) + ) ( 6(x y) ) ( + (x y) ) 3 (x y) t( (x y) ). If x y 3, then g t g xx + (x y) t g x for any α. If x y < 3, we need to find α so that this expression will be negative. So, if we choose i.e., if α, then we have α 6(x y) (+(x y) ) 3 ( +(x y) + (x y) g t g xx + g x. t Set α =. Then the cutoff function is g(x, y, t) = e t( + (x y) ) +. So L(w) and (i) is satisfied. ) ;

8 58 C. CELIK EJDE/CONF/5 Now we show that condition (ii) will be satisfied by w with this cutoff function; i.e., we will show that w on the boundary and at t =. Since g(x, x±, t) =, w on the boundary x y = and x. To show that w at t =, it is sufficient to show that Since lim t x+ lim t x+ x w(x, y, t)h(y)dy = x w(x, y, t)h(y)dy for every h. x+ x Ch(x) h(x) (g(x, y, t)k c (x, y, t) K d (x, y, t))h(y)dy where g C, we have w at t =. Hence, by the maximum principle, (i) and (ii) imply that w inside the domain, which implies K d (x, y, t) e t ( + (x y) + )K c (x, y, t) when x y and x. In the following lemma we use the estimate for K d to get an estimate for u L d which is the linear solution of the Dirichlet problem. Lemma.3. For x, ul d (x, t) c e t u L c (x, t) with u d (x, ) = ke k x, where c is a uniform constant independent of k. Proof. To show this estimate for the linear solution of the Dirichlet problem, the main idea is to use the crucial estimate for the Dirichlet kernel in terms of the Cauchy kernel that we proved in Lemma.. The linear solution for the Dirichlet problem is u L d (x, t) = K d (x, y, t)ke k y dy. Using the estimate of Lemma.. for K d (x, y, t), we have u L d (x, t) = x+ x x+ K d (x, y, t)ke k y dy K d (x, y, t)ke k y dy (since x y ) e t ( x + (x y) + )K c (x, y, t)ke k y dy ce t K c (x, y, t)ke k y dy (since x implies ( + (x y) + ) c) = ce t e (x y) t πt ke k y dy

9 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 59 where u = kx +k t = ce = ce = ce x t e t k πt k x t t e + k x k t(+k t) k x t e +k t πt πt e xs kt ( +k t k t )s ds (where s = ky) k k +k ( t k kx )(s t +k t ) ds e k t u + k t u e u du +k t k t (s kx +k t ), u +k = t k t ( k kx ). The above expression is greater than or equal to k x ce t ke +k t π +k t k x ce t ke +k t π +k t c π e t u L c (x, t)c c e t u L c (x, t), +k t ) and u = u e u du if x <, u e u du if x +k t k t ( k where c = c c π, c = min(c, c ), u e u du c, and u e u du c. Now using the estimates in Lemmas. and.3, we prove that for some initial data u L there is no local solution to the Dirichlet problem (.) in L. Lemma.. For any fixed δ >, we have u d L ((,) (,δ)) = with u = k c kke k x where the c k s will be determined later. Proof. For the solution of the Dirichlet problem, for any fixed δ >, u d L ((,) (,δ)) = > > = δ u d (x, t) dx dt δ t δ t δ t Then using Lemma., we get K d (x, y, t s)dx K d (x, y, t s)(u L d (y, s)) 3 dy ds dx dt K d (x, y, t s)(u L d (y, s)) 3 dy ds dx dt (u L d (y, s)) 3 ce (t s) K c (x, y, t s)dx ce (t s) = ce (t s) π e (x y) (t s) dx π(t s) y (t s) y (t s) K d (x, y, t s) dx dy ds dt. e u du (where u = x y (t s) )

10 6 C. CELIK EJDE/CONF/5 ce (t s) π Ce (t s) y (t s) for some constant C. Using Lemma.3 for u L d (u L c (y, s)) 3 k e u du and the fact that c 3 k k3 e 3k y +k s ( + k s), 3/ we have u d L ((,) (,δ)) for some constant C. = δ t δ t δ t δ t δ t k = k C δ (u L d (y, s)) 3 K d (x, y, t s)dx dy ds dt (u L d (y, s)) 3 (Ce (t s) ) dy ds dt Ce (t s) (c e s u L c (y, s)) 3 dy ds dt Cc e t s Cc e t s C δ t (u L c (y, s)) 3 dy ds dt k c 3 k k + k ds dt s c3 k ln( + k t)dt, c 3 k k3 e 3k y +k s dy ds dt ( + k s) 3/ Now as we discussed for the Cauchy problem in previous section, we choose c k = which will imply that k = e j = k (ln k) / j, for j =,,. So for u (x) = j= j ej e (ej )x, using Lemmas.,.3 and. we can prove the following theorem. Theorem.5. For u = j= j e je (ej )x, the Dirichlet problem (.) has no non-negative local mild solution in L (, ). We have also generalized this result to n-dimensional Dirichlet problem which is u t (x, t) = u(x, t) + u(x, t) n u(x, t) in Ω (, T ), u(x, t) = on Ω (, T ), u(x, ) = u (x) in Ω, where Ω = (, ) (, ) n and obtained the following result. (.3)

11 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 6 Theorem.6. For u (x) = j= j e 6+8n nj n e (ej 6+8n n ) x L, the n - dimensional Dirichlet problem has no non-negative local mild solution in L ((, ) (, )). We have also considered the general nonlinearity for Dirichlet boundary value problem in [3] and proved the following result. Corollary.7. The Dirichlet problem u t (x, t) = u(x, t) + f(u) in Ω (, T ), u(x, t) = on Ω (, T ), u(x, ) = u (x) in Ω, (.) where Ω n is any smooth bounded domain and f(u) u n +, has no nonnegative local mild solution for some u, in L. 3. Non-existence of Local Solutions for the Mixed Boundary Condition In this section, we study the same problem with the mixed boundary conditions; i.e., considering the boundary conditions in terms of u and the normal derivative of u, which is denoted by u n. Let ũ be the solution of the one-dimensional mixed boundary value problem, ũ t (x, t) = ũ xx (x, t) + ũ 3 (x, t) in (, ) (, T ), ũ (x, t) + βũ(x, t) = n x = ± in (, T ), ũ(x, ) = u in (, ) (3.) where β is a constant. First we prove the non-existence of local solution for the one-dimensional mixed boundary value problem (3.) and then we generalize this result to the n - dimensional case. The main idea of the proof is to use the comparison principle for the kernels of heat semigroups with Dirichlet and mixed boundary conditions. ecall that one-dimensional Dirichlet problem is u t (x, t) = u xx (x, t) + u 3 (x, t) in (, ) (, T ), u(±, t) =, u(x, ) = u (x) in (, ). (3.) where u (x) = j= j e je (ej )x for which there is no local solution. Let K(t) and K(t) be the heat kernels on (, ) with Dirichlet and mixed boundary conditions respectively. Now we claim that K(t) K(t) on (, ). In fact, let u be any positive initial data, and let u and ũ be the solution to linear heat equation with Dirichlet and mixed boundary conditions respectively. Hence, (ũ u) n (ũ u) t = (ũ u) xx in (, ) (, T ), + β(ũ u) = u, x = ±, t (, T ), n (ũ u)(x, ) = x (, ), and by the maximum principle, we have ũ u; i.e., K(t) K(t) on (, ). (3.3)

12 6 C. CELIK EJDE/CONF/5 Now suppose for the same initial data u (x) = j= j e je (ej )x problem (3.) has a solution ũ C([.T ], L (, )), then ũ(t) = K(t)u + H(ũ)(t) K(t)u + H(ũ)(t), where H(u)(t) = t K(t s)u3 (s)ds and H(u)(t) = t K(t s)u 3 (s)ds. Let ũ(t) = u (t), and define by iteration, It follows by induction that u k+ (t) = K(t)u + H(u k )(t). K(t)u u k+ (t) u k (t). Thus, the u k (t) converge v(t) (by monotone convergence theorem) which must be a solution of the integral equation v(t) = K(t)u + H(v)(t), which is a contradiction that proves the following theorem for the one dimensional mixed boundary value problem. Theorem 3.. For u (x) = j= j e je (ej )x there is no local L solution to the one-dimensional mixed boundary value problem (3.). Since we can generalize the non-existence result of local solution to the n- dimensional Dirichlet problem, we can also obtain the non-existence of local solution for the n-dimensional mixed boundary value problem by the similar comparison argument as above. Theorem 3.. For the n-dimensional mixed boundary value problem, ũ t (x, t) = ũ(x, t) + ũ(x, t) n ũ(x, t) in (, T ) Ω, ũ n + βũ = on (, T ) Ω, ũ(x, ) = u (x) in Ω, (3.) 6+8n n ) x L, there is no non- where Ω n and u (x) = j= negative local solution in L. j e 6+8n nj n e (ej In the previous sections we give the non-existence results of local solution for some initial data u L for the Dirichlet and mixed boundary value problems. However in the next section, we establish the existence of global solution for some u L +ɛ sufficiently small.. Existence of a Global Solution for Small Initial Data In this section, we give sufficient conditions on q and u for the existence of global solution to the Dirichlet problem u t (x, t) = u xx (x, t) + u 3 (x, t) in (, ) (, T ) u(±, t) = in (, T ) u(x, ) = u in (, ). (.) where u L q (, ). Mainly we will prove that if the initial condition u L +ɛ for any fixed ɛ >, then the Dirichlet problem (.) has a global solution in L +ɛ for u +ɛ sufficiently small.

13 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 63 For notational simplicity, we will use the following form of the solution throughout this section; u(t) = e t u + t e (t s) u 3 (s)ds where e t u denotes the linear solution of problem (.). Before proving the global existence result, we give some interpolation inequalities for the linear solution of (.). emark.. Let < p <. Then it is easy to see that for all t, e t : L L with norm M ; i.e., e t φ φ, and e t : L L with norm M = e tγ where γ is the first (negative) eigenvalue of the Laplacian with vanishing Dirichlet boundary condition; i.e., e t φ e tγ φ. Now we recall the following interpolation theorem. Theorem. (Nirenberg []). Let T (t) be a continuous linear mapping of L p into L p with norm M and L q into L q with norm M. Then T (t) is a continuous mapping of L r into L r with the norm M M λ M λ where p r q and r = λ p + λ q. By this theorem, we conclude that e t : L p L p with the norm M M λ λ M with λ = p p. Hence e t φ p e tγ p p φ p for < p <. For p, we can get a similar estimate by the following interpolation argument. e t : L L with norm M = e tγ as above. Note that for all t, e t : L L with norm M by the maximum principle; i.e., e t φ φ. Then again by the interpolation theorem [] of Nirenberg, we have e t : L p L p with the norm M M λ M λ with λ = p ; i.e., e t φ p e tγ p φ p p [, ). First by the following lemma, we get the estimate on u(s) in terms of u +ɛ, which is a crucial estimate to prove the global existence theorem. Lemma.3. u(s) G(M, T ) u +ɛ s (+ɛ), where G(M, T ) : is an increasing function of M and T respectively when u(s) +ɛ M + on [, T ]. For the proof of the above lemma, we use the following local existence theorem by Brezis and Cazenave (Theorem in []). Theorem.. Assume q > n(p )/ (resp. q = n(p )/) and q (resp. q > ), n. Given any u L q, there exist a time T = T (u ) > and a unique function u C([, T ], L q ) with u() = u, which is a classical solution of (.). Moreover, we have smoothing effect and continuous dependence; namely, u v L q + t n/q u v L C u v L q.

14 6 C. CELIK EJDE/CONF/5 for all t (, T ] where T = min(t (u ), T (v )) and C can be estimated in terms of u L q and v L q. Proof of Lemma.3. By replacing q = + ɛ and p = 3 in this theorem, we get an L estimate for the solution u of problem (.), u(s) G(M, T ) u +ɛ s (+ɛ) 6+6ɛ c(m+) +3ɛ T +3ɛ where G(M, T ) = e 3ɛ and c = c (ɛ). Next we prove the following global existence theorem. Theorem.5. For any fixed ɛ >, there is a δ > such that u +ɛ δ implies that u(t) globally exists and u(t) +ɛ δ for t. Proof. Let us choose δ > such that +ɛ ɛ (G(δ, )δ) < γ and e +ɛ ɛ (G(δ,)δ) <. By the local existence theorem that is stated in Lemma.3, there exists T > such that u(t) exists on [, T ] and u(t) +ɛ δ for t [, T ]. Now we will prove the following claims to prove the global existence result. Claim. u(t) +ɛ < δ for t. Claim. u() +ɛ δ. Proof of Claim. Suppose Claim is not true. Then set T = min{t > u(t) +ɛ δ}. We have u( T ) +ɛ = δ and u(t) +ɛ < δ for all t < T with < T. Consequently, for any t [, T ], the emark. in this section indicates that there ɛ is a γ < (γ = γ +ɛ ) such that u(t) +ɛ e γt u +ɛ + e γt u +ɛ + e γt u +ɛ + t t t e (t s) u 3 (s) +ɛ ds e γ(t s) u(s) +ɛ u(s) ds e γ(t s) u(s) +ɛ Bs +ɛ ds. where B = (G(δ, T )) u +ɛ for notational simplicity. Then e γt u(t) +ɛ u +ɛ + B t e γs u(s) +ɛ s +ɛ ds := H(t), where, we denote the right-hand side of the above inequality by H(t). Then H (t) = Be γt u(t) +ɛ t +ɛ BH(t)t +ɛ. Hence H (t) H(t) Bt +ɛ. Integrating from to t yields which implies ln H(t) H() + ɛ Bt ɛ +ɛ and H(t) H()e +ɛ ɛ Bt +ɛ ɛ, ɛ u(t) +ɛ e γt H(t) u +ɛ e γt e +ɛ ɛ Bt ɛ +ɛ. (.)

15 EJDE-6/CONF/5 EXISTENCE AND NON-EXISTENCE ESULTS 65 Note that for t [, T ] [, ], e γt e +ɛ ɛ Bt ɛ +ɛ e +ɛ ɛ B e +ɛ ɛ (G(δ,)) δ <, by the choice of δ. Then by (.), u( T ) +ɛ < δ which is contrary to the fact that u( T ) +ɛ = δ. So the proof of Claim is completed. Proof of Claim. After proving Claim, Claim follows immediately by the inequality (.) since +ɛ γ+ u() +ɛ δe ɛ B δ by the choice of δ. Finally, the existence of a global solution will follow by a simple induction argument. By Claim and, we can easily conclude that u(j) +ɛ δ for each j =,,.... Using u(j) as new initial data, we can solve the problem for t [j, j + ], and we get u(t) +ɛ δ for all t [j, j + ] by Claim. As a conclusion, we can state the generalized global existence result as follows. Corollary.6. For q >, problem (.) has a global solution in L q for all u L q with u q sufficiently small. eferences [] P. Baras, Non-unicte des solutions d une equation d evolution non-lineaire, Ann. Fac. Sci. Toulouse 5, (983), [] H. Brezis and T. Cazenave, A nonlinear heat equations with singular initial data, Journal D Analyse Mathematique, 68, (996), [3] C. Celik, Z. Zhou, No Local L Solution for a Nonlinear Heat Equation., Communications in PDE s, 8, No. & (3), [] H. Fujita, On the blowing up of the Cauchy problem for u t = u + u +α, J. Fac. Sci. Univ. Tokya, Sect. I, Vol. 3, (966), 9. [5] H. Fujita, On the nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math., Part I, Amer. Math. Soc., 8, (968), [6] Y. Giga, Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 6, (986), 86-. [7] A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J, 3, (98), [8] K. Hayakawa, On nonexistence of global solutions of some nonlinear parabolic equations, Proc. Japan Acad., 9, (973), [9] K. Kobayashi, T. Sirao, and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 9, (977), 7-. [] W. M. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc., 87, (985), [] Nirenberg, L. Functional Analysis, Courant Institute of Mathematical Sciences, New York, [] F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel Journal of Math, 38, No -, (98), 9-. [3] F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal., 3, (979), [] F. B. Weissler, Local existence and non-lexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J., 9, (98), 79. Canan Celik Departmane to Mathematics, TOBB Economics and Technology University, Sögütözü cad. No Sögütözü, Ankara, Turkey address:

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### Pacific Journal of Mathematics

Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation

7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### College of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions

College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use

### Solutions series for some non-harmonic motion equations

Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 115 122. http://ejde.math.swt.edu or http://ejde.math.unt.edu

### m = 4 m = 5 x x x t = 0 u = x 2 + 2t t < 0 t > 1 t = 0 u = x 3 + 6tx t > 1 t < 0

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 21, Number 2, Spring 1991 NODAL PROPERTIES OF SOLUTIONS OF PARABOLIC EQUATIONS SIGURD ANGENENT * 1. Introduction. In this note we review the known facts about

### ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS

ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute

### Extremal equilibria for reaction diffusion equations in bounded domains and applications.

Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,

### 20 Applications of Fourier transform to differential equations

20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use here

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### An Introduction to Partial Differential Equations

An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY

Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,

### TRIPLE FIXED POINTS IN ORDERED METRIC SPACES

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197-207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### MATH PROBLEMS, WITH SOLUTIONS

MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### 1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition

MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification

### Some Problems of Second-Order Rational Difference Equations with Quadratic Terms

International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of Second-Order Rational Difference Equations with Quadratic

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### 5.4 The Heat Equation and Convection-Diffusion

5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The

### 0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup

456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),

### BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

### NON-EXISTENCE AND SYMMETRY OF SOLUTIONS TO THE SCALAR CURVATURE EQUATION. Gabriele Bianchi

NON-EXISTENCE AND SYMMETRY OF SOLUTIONS TO THE SCALAR CURVATURE EQUATION Gabriele Bianchi Dipartimento di Matematica, Universita di Ferrara, Via Machiavelli 35, Ferrara, Italy 0. Introduction The problem

### About the Ingham type proof for the boundary observability of a square and its generalization in N-d

About the Ingham type proof for the boundary observability of a square and its generalization in N-d M. Mehrenberger University of Strasbourg (France), INRIA TONUS team Rome 015 Rome, September 30, 015

### ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE. 1. Introduction

SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties

### SOLUTION TO SECOND-ORDER DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDE

Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 221, pp. 1 6. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION TO SECOND-ORDER

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### Class Meeting # 1: Introduction to PDEs

MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction

DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS

Scientiae Mathematicae Japonicae Online, Vol. 5,, 9 9 9 THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### Lecture 3: Fourier Series: pointwise and uniform convergence.

Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### ( ) ( ) [2],[3],[4],[5],[6],[7]

( ) ( ) Lebesgue Takagi, - []., [2],[3],[4],[5],[6],[7].,. [] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., (984), 83 99. [2] T. Sekiguchi and Y. Shiota, A

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION?

WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION? DAVID M MASON 1 Statistics Program, University of Delaware Newark, DE 19717 email: davidm@udeledu JOEL ZINN 2 Department of Mathematics,

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### ON SUPERCYCLICITY CRITERIA. Nuha H. Hamada Business Administration College Al Ain University of Science and Technology 5-th st, Abu Dhabi, 112612, UAE

International Journal of Pure and Applied Mathematics Volume 101 No. 3 2015, 401-405 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i3.7

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

### Some ergodic theorems of linear systems of interacting diffusions

Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ liuyong@math.pku.edu.cn yangfx@math.pku.edu.cn 1 1 30 1 The ergodic theory of interacting systems has

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### An Introduction to Partial Differential Equations in the Undergraduate Curriculum

An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates

### BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

### FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS

FINITE DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS-SHEN S UNIMODULARITY CONJECTURE AARON TIKUISIS Abstract. It is shown that, for any field F R, any ordered vector space structure

### Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### THE DIFFUSION EQUATION

THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### Existence de solutions à support compact pour un problème elliptique quasilinéaire

Existence de solutions à support compact pour un problème elliptique quasilinéaire Jacques Giacomoni, Habib Mâagli, Paul Sauvy Université de Pau et des Pays de l Adour, L.M.A.P. Faculté de sciences de

### The Australian Journal of Mathematical Analysis and Applications

The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 1-14, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of Matrix-Valued Functions of a Real Variable

### The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

### Convergence and stability results for a class of asymptotically quasi-nonexpansive mappings in the intermediate sense

Functional Analysis, Approximation and Computation 7 1) 2015), 57 66 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac Convergence and

### UNIVERSITETET I OSLO

NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:

### where B.C. means either Dirichlet boundary conditions or Neumann boundary conditions.

Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 23, 2004, 203 212 BIFURCATION OF SOLUTIONS OF ELLIPTIC PROBLEMS: LOCAL AND GLOBAL BEHAVIOUR José L.Gámez Juan F.

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1

MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES JONATHAN CHAPPELON, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract Let K [k,t] be the complete graph on k vertices from which a set of edges,

### NONLOCAL PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS

NONLOCAL PROBLEMS WITH NEUMANN BOUNDARY CONDITIONS SERENA DIPIERRO, XAVIER ROS-OTON, AND ENRICO VALDINOCI Abstract. We introduce a new Neumann problem for the fractional Laplacian arising from a simple

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### A new continuous dependence result for impulsive retarded functional differential equations

CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas

### Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.

Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila

### CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Multiple Solutions to a Boundary Value Problem for an n-th Order Nonlinear Difference Equation

Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp 129 136. ISSN: 1072-6691.

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### THE EIGENVALUES OF THE LAPLACIAN ON DOMAINS WITH SMALL SLITS

THE EIGENVALUES OF THE LAPLACIAN ON DOMAINS WITH SMALL SLITS LUC HILLAIRET AND CHRIS JUDGE Abstract. We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of

### Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial