HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS. 1. Introduction

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1 HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS CARMEN CORTAZAR, MANUEL ELGUETA, ULIO D. ROSSI, AND NOEMI WOLANSKI Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We stu the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions. 1. Introduction The purpose of this article is to show that the solutions of the usual Neumann boundary value problem for the heat equation can be approximated by solutions of a sequence of nonlocal Neumann boundary value problems. Let : R N R be a nonnegative, radial, continuous function with z dz = 1. R N Assume also that is strictly positive in B, d and vanishes in R N \ B, d. Nonlocal evolution equations of the form 1.1 u t x, t = u ux, t = x yuy, t ux, t, R N and variations of it, have been recently widely used to model diffusion processes. More precisely, as stated in [1], if ux, t is thought of as a density at the point x at time t and x y is thought of as the probability distribution of jumping from location y to location x, then R N y xuy, t = ux, t is the rate at which individuals are arriving at position x from all other places and ux, t = R N y xux, t is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation 1.1. For recent references on nonlocal diffusion see, [1], [2], [3], [4], [5], [6], [9], [1], [12], [13], [14] and references therein. Given a bounded, connected and smooth domain, one of the most common boundary conditions that has been imposed in the literature to the heat equation, u t = u, is the Neumann boundary condition, u/ ηx, t = gx, t, x, which leads to the following Key words and phrases. Nonlocal diffusion, boundary value problems. 2 Mathematics Subject Classification 35K57, 35B4. 1

2 2 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI classical problem, 1.2 u t u = in, T, u = g on, T, η ux, = u x in. In this article we propose a nonlocal Neumann boundary value problem, namely 1.3 u t x, t = x y uy, t ux, t + Gx, x ygy, t, R N \ where Gx, ξ is smooth and compactly supported in ξ uniformly in x. In this model the first integral takes into account the diffusion inside. In fact, as we have explained, the integral x yuy, t ux, t takes into account the individuals arriving or leaving position x from or to other places. Since we are integrating in, we are imposing that diffusion takes place only in. The last term takes into account the prescribed flux of individuals that enter or leave the domain. The nonlocal Neumann model 1.3 and the Neumann problem for the heat equation 1.2 share many properties. For example, a comparison principle holds for both equations when G is nonnegative and the asymptotic behavior of their solutions as t is similar, see [8]. Existence and uniqueness of solutions of 1.3 with general G is proved by a fixed point argument in Section 2. Also, a comparison principle when G is proved in that section. Our main goal is to show that the Neumann problem for the heat equation 1.2, can be approximated by suitable nonlocal Neumann problems 1.3. More precisely, for given and G we consider the rescaled kernels 1 ξ 1.4 ξ = C 1 1, G N x, ξ = C 1 G x, ξ N with C1 1 = 1 zzn 2 dz, 2 B,d which is a normalizing constant in order to obtain the Laplacian in the limit instead of a multiple of it. Then, we consider the solution u x, t to u tx, t = 1 2 x yu y, t u x, t G x, x ygy, t, R N \ u x, = u x. We prove in this paper that u u in different topologies according to two different choices of the kernel G.

3 NONLOCAL DIFFUSION PROBLEMS 3 Let us give an heuristic idea in one space dimension, with =, 1, of why the scaling involved in 1.4 is the correct one. We assume that 1 G1, 1 y = G, y = 1 y y and, as stated above, Gx, has compact support independent of x. In this case 1.5 reads u t x, t = x y uy, t ux, t + 1 G x, x y gy, t G x, x y gy, t := A ux, t. If x, 1 a Taylor expansion gives that for any fixed smooth u and small enough, the right hand side A u in 1.5 becomes and if x = and small, A ux = x y uy ux u xx x A u = y uy u + 1 G, y gy C 2 u x g. Analogously, A u1 C 2 / u x 1 + g1. However, the proofs of our results are much more involved than simple Taylor expansions due to the fact that for each > there are points x for which the ball in which integration takes place, Bx, d, is not contained in. Moreover, when working in several space dimensions, one has to take into account the geometry of the domain. Our first result deals with homogeneous boundary conditions, this is, g. Theorem 1.1. Assume g. Let be a bounded C 2+α domain for some < α < 1. Let u C 2+α,1+α/2 [, T ] be the solution to 1.2 and let u be the solution to 1.5 with as above. Then, as. sup u, t u, t L t [,T ] Note that this result holds for every G since g, and that the assumed regularity in u is guaranteed if u C 2+α and u / η =. See, for instance, [11]. We will prove Theorem 1.1 by constructing adequate super and subsolutions and then using comparison arguments to get bounds for the difference u u. Now we will make explicit the functions G we will deal with in the case g. To define the first one let us introduce some notation. As before, let be a bounded C 2+α domain. For x := {x distx, < d} and small enough we write

4 4 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI x = x s d η x where x is the orthogonal projection of x on, < s < and η x is the unit exterior normal to at x. Under these assumptions we define 1.6 G 1 x, ξ = ξ η x ξ for x. Notice that the last integral in 1.5 only involves points x since when y, x y supp implies that x. Hence the above definition makes sense for small. For this choice of the kernel, G = G 1, we have the following result. Theorem 1.2. Let be a bounded C 2+α domain, g C 1+α,1+α/2 R N \ [, T ], u C 2+α,1+α/2 [, T ] the solution to 1.2, for some < α < 1. Let as before and Gx, ξ = G 1 x, ξ, where G 1 is defined by 1.6. Let u be the solution to 1.5. Then, as. sup u, t u, t L 1 t [,T ] Observe that G 1 may fail to be nonnegative and hence a comparison principle may not hold. However, in this case our proof of convergence to the solution of the heat equation does not rely on comparison arguments for 1.3. If we want a nonnegative kernel G, in order to have a comparison principle, we can modify G 1 by taking instead. G 1 x, ξ = G 1 x, ξ + κ ξ = 1 ξ η x ξ + κ 2 Note that for x and y R N \, G 1 x, x y = 1 x y η x x y + κ 2 is nonnegative for small if we choose the constant κ as a bound for the curvature of, since x y d. As will be seen in Remark 4.1, Theorem 1.2 remains valid with G 1 replaced by G 1. Finally, the other Neumann kernel we propose is where C 2 is such that 1.7 d Gx, ξ = G 2 x, ξ = C 2 ξ, {z N >s} z C 2 z N dz ds =. This choice of G is natural since we are considering a flux with a jumping probability that is a scalar multiple of the same jumping probability that moves things in the interior of the domain,. Several properties of solutions to 1.3 have been recently investigated in [8] in the case G = G 2 for different choices of g. For the case of G 2 we can still prove convergence but in a weaker sense. Theorem 1.3. Let be a bounded C 2+α domain, g C 1+α,1+α/2 R N \ [, T ], u C 2+α,1+α/2 [, T ] the solution to 1.2, for some < α < 1. Let as before and

5 NONLOCAL DIFFUSION PROBLEMS 5 Gx, ξ = G 2 x, ξ = C 2 ξ, where C 2 is defined by 1.7. Let u be the solution to 1.5. Then, for each t [, T ] u x, t ux, t weakly in L as. The rest of the paper is organized as follows: in Section 2 we prove existence, uniqueness and a comparison principle for our nonlocal equation. In Section 3 we prove the uniform convergence when g =. In Section 4 we deal with the case G = G 1 and finally in Section 5 we prove our result when G = G Existence and uniqueness In this section we deal with existence and uniqueness of solutions of 1.3. Our result is valid in a general L 1 setting. Theorem 2.1. Let be a bounded domain. Let L 1 R N and G L R N. For every u L 1 and g L loc [, ; L1 R N \ there exists a unique solution u of 1.3 such that u C[, ; L 1 and ux, = u x. As in [7] and [8], existence and uniqueness will be a consequence of Banach s fixed point theorem. We follow closely the ideas of those works in our proof, so we will only outline the main arguments. Fix t > and consider the Banach space X t = C[, t ]; L 1 with the norm w = max t t w, t L 1. We will obtain the solution as a fixed point of the operator T u,g : X t X t t T u,gwx, t = u x + x y wy, s wx, s ds t R N \ Gx, x ygy, t ds. defined by The following lemma is the main ingredient in the proof of existence. Lemma 2.1. Let and G as in Theorem 2.1. Let g, h L, t ; L 1 R N \ and u, v L 1. There exists a constant C depending only on, and G such that for w, z X t, 2.2 T u,gw T v,hz u v L 1 +Ct w z + g h L,t ;L 1 R N \.

6 6 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI Proof. We have T u,gwx, t T v,hzx, t dx u x v x dx t [ ] + x y wy, s zy, s wx, s zx, s ds dx + t R N \ Gx, x y gy, s hy, s ds dx. Therefore, we obtain 2.2. Proof of Theorem 2.1. Let T = T u,g. We check first that T maps X t into X t. From 2.1 we see that for t 1 < t 2 t, t2 t2 T wt 2 T wt 1 L 1 A wy, s ds + B gy, s ds. On the other hand, again from 2.1 t 1 t 1 R N \ T wt u L 1 Ct { w + g L,t ;L 1 R N \}. These two estimates give that T w C[, t ]; L 1. Hence T maps X t into X t. Choose t such that Ct < 1. From Lemma 2.1 we get that T is a strict contraction in X t and the existence and uniqueness part of the theorem follows from Banach s fixed point theorem in the interval [, t ]. To extend the solution to [, we may take as initial datum ux, t L 1 and obtain a solution in [, 2 t ]. Iterating this procedure we get a solution defined in [,. Our next aim is to prove a comparison principle for 1.3 when, G. To this end we define what we understand by sub and supersolutions. Definition 2.1. A function u C[, T ; L 1 is a supersolution of 1.3 if ux, u x and u t x, t x y uy, t ux, t + Gx, x ygy, t. R N \ Subsolutions are defined analogously by reversing the inequalities. Lemma 2.2. Let, G, u and g. If u C [, T ] is a supersolution to 1.3, then u. Proof. Assume that ux, t is negative somewhere. Let vx, t = ux, t + t with so small such that v is still negative somewhere. Then, if we take x, t a point where v

7 NONLOCAL DIFFUSION PROBLEMS 7 attains its negative minimum, there holds that t > and v t x, t = u t x, t + > x yuy, t ux, t = x yvy, t vx, t which is a contradiction. Thus, u. Corollary 2.1. Let, G and bounded. Let u and v in L 1 with u v and g, h L, T ; L 1 R N \ with g h. Let u be a solution of 1.3 with initial condition u and flux g and v be a solution of 1.3 with initial condition v and flux h. Then, u v a.e. Proof. Let w = u v. Then, w is a supersolution with initial datum u v and boundary datum g h. Using the continuity of solutions with respect to the initial and Neumann data Lemma 2.1 and the fact that L R N, G L R N we may assume that u, v C [, T ]. By Lemma 2.2 we obtain that w = u v. So the corollary is proved. Corollary 2.2. Let, G and bounded. Let u C [, T ] resp. v be a supersolution resp. subsolution of 1.3. Then, u v. Proof. It follows the lines of the proof of the previous corollary. 3. Uniform convergence in the case g In order to prove Theorem 1.1 we set w = u u and let ũ be a C 2+α,1+α/2 extension of u to R N [, T ]. We define L v = 1 2 x y vy, t vx, t and L v = 1 2 R N x y vy, t vx, t. Or Then wt = L u u + 1 G x, x ygy, t R N \ = L w + L ũ u + 1 G x, x ygy, t R N \ 1 2 x y ũy, t ũx, t. R N \ w t L w = F x, t,

8 8 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI where, noting that u = ũ in, F x, t = L ũ ũ R N \ R N \ G x, x ygy, t x y ũy, t ũx, t. Our main task in order to prove the uniform convergence result is to get bounds on F. First, we observe that it is well known that by the choice of C 1, the fact that is radially symmetric and ũ C 2+α,1+α/2 R N [, T ], we have that 3.1 sup L ũ ũ L = O α. t [,T ] In fact, C 1 N+2 R N x y ũy, t ũx, t ũx, t becomes, under the change variables z = x y/, C 1 z ũx z, t ũx, t ũx, t 2 R N and hence 3.1 follows by a simple Taylor expansion. Next, we will estimate the last integral in F. We remark that the next lemma is valid for any smooth function, not only for a solution to the heat equation. Lemma 3.1. If θ is a C 2+α,1+α/2 function on R N [, T ] and θ = h on, then for η x = {z dist z, < d} and small, 1 2 x y θy, t θx, t = 1 y x x yη x h x, t R N \ R N \ + x y D β θ [ 2 x, t y x β x x ] β + O α, R N \ β =2 where x is the orthogonal projection of x on the boundary of so that x y 2d. Proof. Since θ C 2+α,1+α/2 R N [, T ] we have θy, t θx, t = θy, t θ x, t θx, t θ x, t = θ x, t y x + β =2 + O x x 2+α + O x y 2+α. D β θ 2 x, t[ y x β x x β]

9 Therefore, 1 2 x y θy, t θx, t = 1 R N \ + x y D β θ [ 2 x, t y x R N \ NONLOCAL DIFFUSION PROBLEMS 9 β =2 R N \ x y θ x, t β x x y x β ] + O α. Fix x. Let us take a new coordinate system such that η x = e N. Since θ η = h on, we get y x x y θ x, t R N \ = R N \ x yη x y x h x, t + R N \ x y θ xi x, t y i x i We will estimate this last integral. Since is a C 2+α domain we can chose vectors e 1, e 2,..., e so that there exists κ > and constants f i x such that { B 2d x Therefore x y R N \ = y N x N + f i xy i x i 2 > κ 2+α } R N \, { B 2d x y N x } N + f i xy i x i 2 < κ 2+α. R N \ + = I 1 + I 2. θ xi x, t y i x i yn x N + f i xy i x i 2 κ 2+α x y y N x N + x y f i xy i x i 2 >κ 2+α. θ xi x, t y i x i θ xi x, t y i x i If we take z = y x/ as a new variable, recalling that x N x N = s, we obtain I 1 C 1 θ xi x, t zn s+ f i xz i 2 κ 1+α z z i dz C κ 1+α.

10 1 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI On the other hand, I 2 = C 1 θ xi x, t z N s+ f i xz i 2 >κ 1+α z z i dz. Fix 1 i N 1. Then, since { is radially symmetric, z z i is an odd function of the variable z i and, since the set z N s + f i xz i 2 } > κ 1+α is symmetric in that variable we get I 2 =. Collecting the previous estimates the lemma is proved. We will also need the following inequality. Lemma 3.2. There exist K > and > such that, for <, y x 3.2 x yη x K x y. R N \ R N \ Proof. Let us put the origin at the point x and take a coordinate system such that η x = e N. Then, x =, µ with < µ < d. Then, arguing as before, y x x yη x = x y y N + µ = {y N >κ 2 } {y N >κ 2 } Fix c 1 small such that 1 2 x y y N + µ x y y N + µ {z N >} + C. R N \ z z N dz 2c 1 R N \ R N \ { y N <κ 2 } {<z N <2c 1 } x y y N + µ z dz. We divide our arguments into two cases according to whether µ c 1 or µ > c 1. Case I Assume µ c 1. In this case we have, x y y N + µ = C 1 z z N dz {y N >κ 2 } {z N >κ+ µ } 3.3 = C 1 z z N dz z z N dz {z N >} {<z N <κ+ µ } C 1 z z N dz 2c 1 z dz C 1 2 {z N >} {<z N <2c 1 } {z N >} z z N dz.

11 Then, R N \ C if is small enough and NONLOCAL DIFFUSION PROBLEMS 11 x yη x {z N >} y x K R N \ z z N dz K C, K < 1 4 {z N >} z z N dz. x y Case II Assume that µ c 1. For y in R N \ B x, d we have y N κ. Then, x y y N + µ K x y R N \ R N \ c 1 κ x y K x y R N \ R N \ = c 1 κ K R N \ x y, if is small and This ends the proof of 3.2. We now prove Theorem 1.1. K < c 1 2. Proof of Theorem 1.1. We will use a comparison argument. First, let us look for a supersolution. Let us pick an auxiliary function v as a solution to v t v = hx, t in, T, v η = g 1x, t on, T, vx, = v 1 x in. for some smooth functions hx, t 1, g 1 x, t 1 and v 1 x such that the resulting v has an extension ṽ that belongs to C 2+α,1+α/2 R N [, T ], and let M be an upper bound for v in [, T ]. Then, v t = L v + v L ṽ x yṽy, t ṽx, t + hx, t. R N \

12 12 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI Since v = ṽ in, we have that v is a solution to { v t L v = Hx, t, in, T, vx, = v 1 x in, where by 3.1, Lemma 3.1 and the fact that h 1, Hx, t, = ṽ L ṽ x yṽy, t ṽx, t + hx, t 1 + R N \ R N \ g1 x, t R N \ x yη x x y R N \ β =2 x yη x y x g 1 x, t D β ṽ [ 2 x, t y x y x for some constant D 1 if is small so that C α 1/2. β x x ] β + 1 C α D 1 R N \ x y Now, observe that Lemma 3.2 implies that for every constant C > there exists such that, 1 y x x yη x C x y, R N \ R N \ if <. Now, since g =, by 3.1 and Lemma 3.1 we obtain F C α + x y R N \ C α + C 2 R N \ β =2 x y. D β ũ [ 2 x, t y x Given δ >, let v δ = δv. Then v δ verifies { v δ t L v δ = δhx, t, in, T, v δ x, = δv 1 x in. β x x ] β By our previous estimates, there exists = δ such that for, F δhx, t,. So, by the comparison principle for any it holds that Therefore, for every δ >, Mδ v δ w v δ Mδ. Mδ lim inf w lim sup w Mδ.

13 NONLOCAL DIFFUSION PROBLEMS 13 and the theorem is proved. 4. Convergence in L 1 in the case G = G 1 First we prove that F goes to zero as goes to zero. Lemma 4.1. If G = G 1 then as. R N \ F x, t in L [, T ]; L 1 Proof. As G = G 1 = ξ η x ξ, for x, by 3.1 and Lemma 3.1, F x, t = 1 y x x yη x gy, t g x, t R N \ x y D β ũ [ 2 x, t y x β x x ] β + O α. β =2 As g is smooth, we have that F is bounded in. Recalling the fact that = O and F x, t = O α on \ we get the convergence result. We are now rea to prove Theorem 1.2. Proof of Theorem 1.2. In the case G = G 1 we have proven in Lemma 4.1 that F in L 1 [, T ]. On the other hand, we have that w = u u is a solution to Let z be a solution to Then z is a solution to By comparison we have that w t L w = F wx, =. z t L z = F zx, =. z t L z = F zx, =. z w z and z. Integrating the equation for z we get z, t L 1 = z x, t dx = Applying Lemma 4.1 we get as. So the theorem is proved. t sup z, t L 1 t [,T ] F x, s ds dx.

14 14 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI Remark 4.1. Notice that if we consider a kernel G which is a modification of G 1 of the form with G x, ξ = G 1 x, ξ + Ax, ξ, R N \ Ax, x y, in L 1 as, then the conclusion of Theorem 1.2 is still valid. In particular, we can take Ax, ξ, = κ ξ. 5. Weak convergence in L 1 in the case G = G 2 First, we prove that in this case F goes to zero as measures. Lemma 5.1. If G = G 2 then there exists a constant C independent of such that T F x, s dx ds C. Moreover, F x, t as measures as. That is, for any continuous function θ, it holds that T F x, tθx, t dx dt as. Proof. As G = G 2 = C 2 ξ and g and ũ are smooth, taking again the coordinate system of Lemma 3.1, we obtain F x, t = 1 x y C 2 gy, t y N x N g x, t R N \ 1 x y ũ xi x, t y i x i R N \ x y D β ũ x, t [ y x β x x ] β + O α R N \ 2 β =2 = 1 x y C 2 g x, t y N x N g x, t R N \ 1 x y ũ xi x, t y i x i + O1χ + O α. R N \

15 Let NONLOCAL DIFFUSION PROBLEMS 15 B x, t := x y C 2 g x, t y N x N R N \ R N \ x y ũ xi x, t y i x i g x, t Proceeding in a similar way as in the proof of Lemma 3.1 we get for small, x y C 2 g x, t y N x N g x, t R N \ = g x, t x y C 2 y N x N And + g x, t g x, t R N \ { y N x N κ 2 } R N \ {y N x N >} R N \ {<y N x N <κ 2 }. x y C 2 y N x N = C 1 g x, t zc 2 z N dz + Oχ. {z N >s} = R N \ + x y ũ xi x, t y i x i ũ xi x, t ũ xi x, t = C 1 ũ xi x, t = I 2 + Oχ. { y N x N κ 2 } {y N x N >κ 2 } {z N s>κ} x y C 2 y N x N x y y i x i x y y i x i zz i dz + Oχ As in Lemma 3.1 we have I 2 =. Therefore, B x, t = C 1 g x, t z C 2 z N dz + Oχ. {z N >s} Now, we observe that B is bounded and supported in. Hence t F x, τ dx dτ 1 t B x, τ dx dτ + Ct + Ct α C. This proves the first assertion of the lemma.

16 16 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI Now, let us write for a point x x = x µη x with < µ < d. For small and < µ < d, let ds µ be the area element of {x dist x, = µ}. Then, ds µ = ds + O, where ds is the area element of. So that, taking now µ = s we get for any continuous test function θ, 1 T B x, tθ x, t dx dt T d = O + C 1 g x, tθ x, t z C 2 z N dz ds ds dt = O as, since we have chosen C 2 so that d {z N >s} {z N >s} z C 2 z N dz ds =. Now, with all these estimates, we go back to F. We have F x, t = 1 B x, t + O1χ + O α. Thus, we obtain T F x, tθ x, t dx dt as. Now, if σr is the modulus of continuity of θ, T T F x, tθx, t dx dt = F x, tθ x, t dx dt T + F x, t θx, t θ x, t dx dt T T F x, tθ x, t dx dt + Cσ F x, t dx dt as. Finally, the observation that F = O α in \ gives T and this ends the proof. \ F x, tθx, t dx dt as Now we prove that u is uniformly bounded when G = G 2. Lemma 5.2. Let G = G 2. There exists a constant C independent of such that u L [,T ] C.

17 NONLOCAL DIFFUSION PROBLEMS 17 Proof. Again we will use a comparison argument. Let us look for a supersolution. Pick an auxiliary function v as a solution to v t v = hx, t in, T, 5.1 v η = g 1x, t on, T, vx, = v 1 x in. for some smooth functions hx, t 1, v 1 x u x and g 1 x, t 2 K C max gx, t + 1 K as in 3.2 [,T ] such that the resulting v has an extension ṽ that belongs to C 2+α,1+α/2 R N [, T ] and let M be an upper bound for v in [, T ]. As before v is a solution to { v t L v = Hx, t, in, T, vx, = v 1 x in, where H verifies g1 x, t Hx, t, So that, by Lemma 3.2, for <. R N \ x yη x Hx, t, g 1 x, t K Let us recall that F x, t = L ũ ũ + C y x D 1 R N \ D 1 x y + 1 R N \ 2 R N \ R N \ x ygy, t Then, proceeding once again as in Lemma 3.1 we have, F x, t g x, t C 2 g x, t x y + R N \ + C α + C x y R N \ [ C2 + 1 ] max gx, t + C [,T ] g 1 x, t K 2 if <, by our choice of g 1. x y ũy, t ũx, t. R N \ + C x y + C α R N \ R N \ x y + C α x y x y y x η x

18 18 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI Therefore, for every small enough, we obtain F x, t Hx, t,, and, by a comparison argument, we conclude that M vx, t u x, t vx, t M, for every x, t [, T ]. This ends the proof. Finally, we prove our last result, Theorem 1.3. Proof of Theorem 1.3. By Lemma 5.1 we have that as. F x, t as measures in [, T ] Assume first that ψ C 2+α and let ϕ be the solution to w t L w = wx, = ψx. Let ϕ be a solution to ϕ t ϕ = ϕ η = ϕx, = ψx. Then, by Theorem 1.1 we know that ϕ ϕ uniformly in [, T ]. For a fixed t > set ϕ x, s = ϕ x, t s. Then ϕ satisfies ϕ s + L ϕ =, for s < t, ϕx, t = ψx. Analogously, set ϕx, s = ϕx, t s. Then ϕ satisfies ϕ t + ϕ = ϕ η = ϕx, t = ψx.

19 NONLOCAL DIFFUSION PROBLEMS 19 Then, for w = u u we have t w w t x, t ψx dx = s x, s ϕ ϕ x, s dx ds + s x, s w x, s dx ds t t = L w x, sϕ x, s dx ds + F x, s ϕ x, s dx ds t ϕ + s x, s w x, s dx ds t t = L ϕ x, sw x, s dx ds + F x, s ϕ x, s dx ds t ϕ + s x, s w x, s dx ds t = F x, sϕ x, s dx ds. Now we observe that, by the Lemma 5.1, t t F x, sϕ x, s dx ds + sup ϕ x, s ϕx, s L <s<t F x, sϕx, s dx ds t F x, s dx ds as. This proves the result when ψ C 2+α. Now we deal with the general case. Let ψ L 1. Choose ψ n C 2+α such that ψ n ψ in L 1. We have w x, t ψx dx w x, t ψ n x dx + ψ n ψ L 1 w L. By Lemma 5.2, {w } is uniformly bounded, and hence the result follows. Acknowledgements. Supported by Universidad de Buenos Aires under grants X52 and X66, by ANPCyT PICT No , Fundacion Antorchas Project 139-5, by CONICET Argentina and by FONDECYT Project and Coop. Int Chile. References [1] P. Bates and A. Chmaj. An integrodifferential model for phase transitions: stationary solutions in higher dimensions.. Statistical Phys., 95, , [2] P. Bates and A. Chmaj. A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal., 15, , [3] P. Bates, P. Fife, X. Ren and X. Wang. Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal., 138, , 1997.

20 2 C. CORTAZAR, M. ELGUETA,.D. ROSSI, AND N. WOLANSKI [4] P. Bates and. Han. The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation. To appear in. Math. Anal. Appl. [5] P. Bates and. Han. The Neumann boundary problem for a nonlocal Cahn-Hilliard equation.. Differential Equations, 212, , 25. [6] C. Carrillo and P. Fife. Spatial effects in discrete generation population models.. Math. Biol. 52, , 25. [7] C. Cortazar, M. Elgueta and. D. Rossi. A non-local diffusion equation whose solutions develop a free boundary. Ann. Henri Poincare, 62, , 25. [8] C. Cortazar, M. Elgueta,. D. Rossi and N. Wolanski. Boundary fluxes for non-local diffusion. Preprint. [9] X Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, , [1] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, , Springer, Berlin, 23. [11] A. Friedman. Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N, [12] C. Lederman and N. Wolanski. Singular perturbation in a nonlocal diffusion problem. To appear in Comm. Partial Differential Equations. [13] X. Wang. Metaestability and stability of patterns in a convolution model for phase transitions.. Differential Equations, 183, , 22. [14] L. Zhang. Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks.. Differential Equations 1971, , 24. Carmen Cortazar and Manuel Elgueta Departamento de Matemática, Universidad Catolica de Chile, Casilla 36, Correo 22, Santiago, Chile. address: ccortaza@mat.puc.cl, melgueta@mat.puc.cl. ulio D. Rossi Consejo Superior de Investigaciones Científicas CSIC, Serrano 123, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA 1428 Buenos Aires, Argentina. address: jrossi@dm.uba.ar Noemi Wolanski Departamento de Matemática, FCEyN UBA 1428 Buenos Aires, Argentina. address: wolanski@dm.uba.ar