Nonlocal and nonlinear evolution equations in perforated domains

NONLOCAL AND NONLINEAR EVOLUTION EQUATIONS IN PERFORATED DOMAINS MARCONE C. PEREIRA AND SILVIA SASTRE-GOMEZ Abstract. In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form ut(x, t) = RJ(x−y)u(y, t)dy −hε(x)u(x, t) + f(x, u(x, t)) with xin a perturbed domain Ωε⊂Ω which is thought as a fixed set Ω from where we remove a subset Aεcalled the holes. We choose an appropriated families of functions hε∈L∞in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take Jas a non-singular kernel and fas a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ωεhave a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation. 1. Introduction and main results Let Ωε⊂RNbe a family of open bounded sets satisfying Ωε⊂Ω for some fixed open bounded domain Ω ⊂RNand a positive parameter ε. Denoting χε∈L∞(RN) by the characteristic function of Ωε, we assume that there exists a function X ∈ L∞(RN), strictly positive inside Ω such that χε*Xweakly∗in L∞(Ω). More precisely, we suppose (1.1) ZΩ χε(x)ϕ(x)dx →ZΩ X(x)ϕ(x)dx, as ε→0, for all ϕ∈L1(Ω), and there exists a positive constant c > 0 such that (1.2) X(x)≥c > 0 for all x∈Ω. Notice that we also have X(x)≤1 in Ω with X(x)≡0 in RN\Ω since the family of characteristic functions χεsatisfy the same conditions for all ε > 0. Here, we see the family of open sets Ωεas a family of perforated domains where the set Aε= Ω \Ωε can be thought as the holes inside Ω. Our main goal is to analyze the asymptotic behavior of the solutions of a semilinear nonlocal evolution problem with nonlocal reaction and nonsingular kernel in perforated domains as ε→0. We consider problems of the form (1.3) ut(x, t) = ZRN\Aε J(x−y)u(y, t)dy −hε(x)u(x, t) + f(x, u(x, t)), u(x, 0) = u0(x), Key words and phrases. perforated domains, nonlocal equations, semilinear equations, Dirichlet problem. 2010 Mathematics Subject Classification. 45A05, 45M05, 49J40. 1 arXiv:2004.02348v1 [math.AP] 5 Apr 2020 2 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ for x∈Ωεand tin bounded intervals taking nonlinearities f: Ωε×L1(Ωε)7→ Rwhich are nonlocal reaction terms defined by f(x, u)=(g◦mΩε)(x, u) under the following conditions: (Hf) We assume g:R7→ Ris a smooth function, globally Lipschitz, and mΩε: Ωε×L1(Ωε)7→ R is given by mΩε(x, u) = 1 |Bδ(x)∩Ωε|ZBδ(x)∩Ωε u(y)dy where Bδ(x) is the ball of radius δ > 0 centered at x∈Ωε. Here, and along the whole paper, the function Jis a smooth non-singular kernel satisfying (HJ) J∈ C(RN,R) is non-negative with J(0) >0, J(−x) = J(x) for every x∈RNand ZRN J(x)dx = 1. We consider both Dirichlet and Neumann nonlocal problems. For the Dirichlet case we impose hε(x)≡1 with uvanishing in RN\Ωεwhile in the Neumann case we consider hε(x) = ZRN\Aε J(x−y)dy, x ∈Ωε, only assuming that uvanishes in RN\Ω. Note that for the former we have considered nonlocal Neumann boundary conditions in the holes Aεand a Dirichlet boundary condition in the exterior of the set Ω. It is not difficult to see that there are positive constants ε0and C0such that (1.4) |Bδ(x)∩Ω|=ZBδ(x)∩Ωε χε(y)dy ≥C0 for all x∈Ω and 0 < ε < ε0. In particular, we have RBδ(x)X(y)dy ≥C0for all x∈Ω. Indeed, inequality (1.4) follows from (1.1) and (1.2) since Ω ⊂RNis a bounded open set. See also that the map mΩεtransforms (x, u) into mΩε(x, u), the average of function uin a set given by the intersection between the ball Bδ(x) with the perforated domain Ωεsetting the nonlocal nonlinear effect in our model. According to [18], equation (1.3) can be seen as a continuous model for a single species in a finite N-dimensional habitat where the density of the population at position xand time tis given by a function u(x, t). Hostile surroundings are modeled by the Dirichlet conditions whereas the Neumann condition is the standard approach to modeling species in geographically isolated regions. The nonlocal effect under reaction terms is discussed for instance in [13]. It is used to model situations where the total biomass plays a role and the model incorporates group defense or visual communications. See also [9, 14, 20]. In fact, there exists a big interest in the study of nonlocal diffusion equations to model different problems from different areas. We still mention [1, 2, 10, 12, 21, 22] and references NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 3 therein where population dynamical processes and chemical reaction-diffusion models are treated. In [15], an economic model to fluctuation of stock market is presented. The paper is organized as follows: in Section 2, we introduce the main results of the paper discussing the classic situation known as periodic perforated domains. In Section 3, we study existence and uniqueness of the solutions to (1.3) obtaining uniform estimates on parameters εand δ > 0. The proofs of our main results Theorem 2.1 and Theorem 2.2 are given in Section 4. In Section 5, we discuss the asymptotic behavior of the limit equations, when parameter δgoes to zero obtaining a nonlocal problem with local nonlinearity. In this way, we give a scenario taking first ε→0, and next δ→0. In a forthcoming paper, we will study the reversed limit. As noticed in [25, 27], a double limit commuting it is not expected. Finally, we emphasize that the results obtained here are also in agreement with the previous works [26, 27, 28] where nonlocal linear equations have been considered. Therefore, this work is a natural continuation for nonlocal and nonlinear equations in perforated domains. 2. Main results We have the following result for the Dirichlet problem: Theorem 2.1. Let {uε}ε>0be the family of solutions given by problem (1.3) under conditions (2.5) uε(x, t)≡0for all x∈RN\Ωεand t > 0 uε(x, 0) = u0(x)in L2(Ω) and hε(x)≡1in RN. Then, there exists u∗:R×RN7→ Rwith u∗(x, t)≡0in RN\Ωand u∗∈C1([a, b], L2(RN)) for any closed interval [a, b]⊂R, such that, as ε→0, uε* u∗weakly∗in L∞([a, b]; L2(Ω)). Furthermore, we have that limit function u∗satisfies the following nonlocal equation in Ω ut(x, t) = X(x)ZRN J(x−y) (u(y, t)−u(x, t)) dy +X(x)fX(x, u(x, t)) + (X(x)−1) u(x, t) with u(x, t)≡0in x∈RN\Ωand t > 0 u(x, 0) = X(x)u0(x) where (2.8) fX(x, u) = (g◦mX)(x, u) 4 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ and (2.9) mX(x, u) = 1 RBδ(x)X(y)dy ZBδ(x) u(y)dy. Concerning to Neumann conditions on the holes Aεwe have: Theorem 2.2. Let {uε}ε>0be the family of solutions given by (1.3) with (2.10) uε(x, t)≡0for all x∈RN\Ωand t > 0 uε(x, 0) = u0(x)in L2(Ω) and hε(x) = ZRN\Aε J(x−y)dy for x∈RN. Then, there exists u∗:R×RN7→ Rwith u∗(x, t)≡0in RN\Ωand u∗∈ C1([a, b], L2(RN)) for any closed interval [a, b]⊂R, such that, as ε→0, ˜uε* u∗weakly∗in L∞([a, b]; L2(Ω)) where ˜ ·denotes the extension by zero of functions defined on subsets of RN. Furthermore, we have the limit function u∗satisfies the following nonlocal equation in Ω (2.11) ut(x, t) = X(x)ZRN J(x−y) (u(y, t)−u(x, t)) dy +X(x)fX(x, u(x, t)) −Λ(x)u(x, t) with u(x, t)≡0for x∈RN\Ωand t > 0 u(0, x) = X(x)u0(x) where the coefficient Λ∈L∞(RN)is given by Λ(x) = ZRN J(x−y) (1 −χΩ(y) + X(y)) dy − X(x) and fX=g◦mXwith mXdefined by (2.9). We point out the dependence of both limit equations on the term X. They establish the effect of the holes in the original equation (1.3). Indeed, a kind of friction or drag coefficient is obtained, as well as, a new reaction nonlinearity, both caused by the perforations. Also, if we rewrite the more involved term Λ appearing in Theorem 2.2 as Λ(x) = ZRN\Ω J(x−y)dy +ZRN J(x−y) (X(y)− X(x)) dy, x ∈Ω, we see that the kernel Jexplicitly affects the limit equation for the Neumann problem. As we can see, such dependence on the kernel Jdoes not occur in the Dirichlet problem where the coefficient only depends on the perturbations via X. NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 5 Concerning to the extreme case X(x)≡1 in Ω, we can argue as in [28, Corollary 3.1] to see that, if X(x)≡1 in Ω, then the limit equation for both conditions is the nonlocal Dirichlet problem in Ω, namely ut(t, x) = ZRN J(x−y)(u(t, y)−u(t, x))dy +ˆ f(x, u(x, t)), u(0, x) = u0(x), x∈Ω, t ∈R, with u(t, x)≡0 in x∈RN\Ω and ˆ f: Ω ×L1(RN)7→ Rdefined by ˆ f(x, u)=(g◦mBδ)(x, u) where mBδis the average of uon the ball Bδ(x) (2.12) mBδ(x, u) = 1 |Bδ(x)|ZBδ(x) u(y)dy. Hence, we can say that small holes do not make any effect on the limit process. Finally, we notice the degenerated case X(x)≡0 in RNis not considered here, since we work under condition (1.2). It is a subject of a forthcoming paper. See that the solutions of the limit problems given by Theorems 2.1 and 2.2 depend on the parameter δ > 0 which sets the average function mXdefined at (2.9). Here, we also analyse the asymptotic behavior of these equations as δ→0 under the additional conditions J,X and u0of class C1and gof class C2. It is necessary to guarantee strong convergence in L2(Ω) since we do not have regularizing effect for these nonlocal equations. Theorem 2.3. Let {uδ}δ>0be the family of solutions given by (2.13) uδ t(x, t) = X(x)ZRN J(x−y)uδ(y, t)dy −h0(x)uδ(x, t) + X(x)fX(x, uδ(x, t)), uδ(x, t)≡0for x∈RN\Ω uδ(0, x) = X(x)u0(x) under conditions J,Xand u0of class C1,gof class C2, (2.14) uδ(x, t)≡0for all x∈RN\Ωand t > 0 uδ(x, 0) = X(x)u0(x)in L2(Ω) with h0(x)≡1in RNfor the Dirichlet problem and h0(x) = ZRN J(x−y)(1 −χΩ(y) + X(y)) dy for the Neumann problem. Then, there exists ¯u:R×RN7→ Rwith ¯u(x, t)≡0in RN\Ωand ¯u∈C1([a, b], L2(RN)) for any closed interval [a, b]⊂R, such that, uδ*¯uweakly∗in L∞([a, b]; L2(Ω)) as δ→0. 6 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ Furthermore, we have the limit function ¯usatisfies the following nonlocal equation in Ω ut(x, t) = X(x)ZRN J(x−y)u(y, t)dy −h0(x)u(x, t) + X(x)g1 X(x)u(x, t) with u(x, t)≡0in x∈RN\Ωand u(x, 0) = X(x)u0(x). As expected, a local reaction term is obtained at the limit. The effect of the perforations can be seen, and a nontrivial term is captured. As in the previous results, under small perforations, that is, assuming X(x)≡1 in Ω, a nonlocal Dirichlet equations in Ω is obtained for both Dirichlet and Neumann problems with a local reaction just set by function g. Under this additional condition Theorem 2.3 implies the following limit equation ut(t, x) = ZRN J(x−y)(u(t, y)−u(t, x))dy +g(u(x, t)), u(0, x) = u0(x), x∈Ω, t ∈R, with u(t, x)≡0 in x∈RN\Ω. Purely periodic perforations. The study of solutions in periodic perforated domains has attracted much interest. For local operators, from pioneering works to recent ones, we may mention [3, 4, 5, 6, 7, 16, 23, 29, 32] and references therein that are concerned with elliptic and parabolic equations, nonlinear operators, as well as Stokes and Navier-Stokes equations from fluid mechanics. For instance, in the classical paper [8], the authors analyze the Dirichlet problem for the Laplacian in a bounded domain from where a big number of periodic small balls are removed. They consider Ωε= Ω \ ∪iBrε(xi) where Brε(xi) is a ball centered in xi∈Ω of the form xi∈2εZNwith radius 0 < rε< ε ≤1. Figure 1 bellow ilustrates a periodic perforated domain Ωε. Ωϵ Figure 1. A periodic perforated domain Ωε= (0,1)2\ ∪Brε(xi). In this periodic case, it is known, see for instance [27, Section 4], that χε*X=|Q\B|/|Q|as ε→0 NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 7 where here, Qdenotes the unit cube and Bis a ball inside the cube. See that Xis a positive constant. Thus, due to Theorems 2.1 and 2.2, we have that the homogenized equations associated to (1.3) under Dirichlet and Neumann conditions are respectively: ρut(x, t) = ZRN J(x−y) (u(y, t)−u(x, t)) dy + (1 −ρ)u(x, t) + g(ρ mBδ(x, u(·, t))) and ρut(x, t) = ZRN J(x−y) (u(y, t)−u(x, t)) dy −ZRN\Ω J(x−y)dy +g(ρ mBδ(x, u(·, t))) for x∈Ω, t∈Rwith u(x, t)≡0 in RN\Ω where ρ=X−1=|Q|/|Q\B| is a strictly positive constant larger than one since |B|>0 and mBδis the average (2.12). Notice that even in this classic and standard situation, the nonlinear term of (1.3) is perturbed in a non trivial way. Indeed, we have fX(·, u) = g(ρ mBδ(·, u)). It is due to the fact that the integral operators considered here do not regularize, and hence solutions uεwith initial conditions in L2are expected to be bounded in L2but nothing better. We can still assume appropriated conditions on J,Xand u0to use Theorem 2.3 and pass to the limit in the previous equations as δ→0 obtaining a local reaction term, also depending on ρand given by g(ρ u(·, t)). 3. Existence and uniform boundedness In this section, we mainly prove existence and uniqueness of the solutions to problem (1.3) giving uniform bounds with respect to parameters εand δ > 0. We also introduce a technical result concerning to the convergence of integral expressions under sequence of functions. Let us consider here B=RN\Aεand u(x)≡0 in x∈RN\Ω for the Neumann problem and B=RNand u(x)≡0 in x∈RN\Ωεfor the Dirichlet problem. Since we are assuming J∈ C(RN,R), we have that the operator (3.16) Ku(x) = ZB J(x−y)u(y)dy. satisfies K:L2(Ω) →L2(Ω). On the other hand, it follows from (1.4) that aε(x) = 1 |Bδ(x)∩Ωε| is uniformly bounded in ε. Indeed, aε∈L∞(Ω) and satisfies (3.17) 0 ≤aε(x)≤1 C0 ,∀x∈Ω and ε∈(0, ε0). 8 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ We are first interested in the Nemitcky operator associated to f, given by F:L2(Ωε)→L2(Ωε) with F(u)(x) = f(x, u) = g(mΩε(x, u)). To study the properties of F, we first see Mu(x) = mΩε(x, u). In the following lemma we state that the Nemitcky operator Massociated to mΩεis continuous, globally Lipschitz and compact. For a proof, see [19]. Lemma 3.1. Let (Ω, µ, d)be a metric measure space with µ(Ω) <∞, and set the operator M(u)(x) = aε(x)ZBδ(x)∩Ωε u(y)dy. Since the function aε∈L∞(Ω) satisfies (3.17), we have that M∈ L(L1(Ω), L∞(Ω)) with M:L2(Ω) →L2(Ω) being a compact operator. Remark 3.1. In particular, if the nonlinear function g:R→Ris globally Lipschitz, then the Nemitcky operator associated to gand set by G:L2(Ω) →L2(Ω) is also globally Lipschitz. Hence, since Mis a bounded operator by Lemma 3.1, we get F=G◦M:L2(Ω) →L2(Ω) globally Lipschitz. Then, there exists a constant C > 0, independent of ε > 0such that kF(u)−F(v)kL2(Ω) ≤Cku−vkL2(Ω). In the following Proposition, we prove the existence and uniqueness of the solutions to (1.3) and we give a uniform bound of uεwith respect to ε > 0. Proposition 3.1. The problem (1.3), under the assumptions uε(x, t)≡0for all x∈RN\Ωand t > 0and hε(x) = ZRN\Aε J(x−y)dy for x∈RN, for the Neumann problem, or uε(x, t)≡0for all x∈RN\Ωεand t > 0and hε(x)≡1in RN, for the Dirichlet problem, has a unique global solution uε: Ω ×R→Rwith uε∈ C1([a, b], L2(Ωε)) for every u0∈L2(Ω) and any bounded interval [a, b]⊂R, with (3.18) uε(·, t) = e−hε(x)tu0+Zt 0 e−hε(x)(t−s)[Ku +F(uε)](·, s)ds ∀t∈R. Moreover, there exists constants αand D, with D > 0, independent of εand t, such that (3.19) kuε(·, t)kL2(Ωε)≤eαt ku0kL2(Ω) +Dt. Proof. The existence and uniqueness result is proved using a fixed point argument with the variations of constants formula on the right of (3.18) in C([−T, T ], L2(Ω)) for some T > 0 independent of the initial data, and with a prolongation argument. Considering the formula on the right of (3.18) as an operator defined from L1([−T, T], L2(Ω)) into C([−T, T], L2(Ω)), NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 9 we have the uniqueness in both spaces and applying Theorem in [24, p. 109] we obtain that uis a strong solution of (1.3) in C1([−T, T], L2(Ω)). To prove the uniform bound, we consider B=RN\Aεfor the Neumann problem and B=RNfor the Dirichlet problem. d dt 1 2kuε(·, t)k2 L2(Ωε)=d dt 1 2ZB (uε(x, t))2dx =ZB uε(x, t)uε t(x, t)dx =ZB uε(x, t)ZB J(x−y)uε(y, t)dy −hε(x)uε(x, t) + f(x, uε(x, t))dx =−1 2ZBZB J(x−y)(uε(y, t)−uε(x, t))2dy dx +ZB uε(x, t)f(x, uε(x, t)) dx Considering λε 1= inf u∈W 1 2ZBZB J(x−y)(uε(y, t)−uε(x, t))2dy dx ZB (uε(x, t))2dx where W={u∈L2(RN\Aε) : u(x)≡0∀x∈RN\Ω}for the Neumann problem, and W={u∈L2(RN) : u(x)≡0∀x∈RN\Ωε}for the Dirichlet problem. Thanks to Young’s inequality and Remark 3.1, since gis globally Lipschitz, we have that d dt 1 2kuε(·, t)k2 L2(Ωε)≤(η2−λε 1)ZB (uε(x, t))2dx +η−2kf(·, uε(·, t))k2 L2(B) ≤(η2−λε 1+η−2C)kuε(·, t)k2 L2(Ωε)+|Ω||g(0)| for any η > 0. Therefore integrating in [0, t], we conclude kuε(·, t)kL2(Ωε)≤e2(η2−λε 1+η−2C)tku0kL2(Ω) +|Ω||g(0)|t finishing the proof.  Remark 3.2. The term λε 1is the first eigenvalue of ZB J(x−y)uε(y)dy −hε(x)uε(x)−λε 1uε(x)=0. From [27], we know that the family λε 1is lower bounded for both the Dirichlet and Neumann problems. For the Neumann problem, it is obtained under the additional condition: There exists finite family of sets B0, B1, . . . , BL⊂RN\Aε such that B0=RN\Ω, RN\Aε⊂ L [ i=0 Biand αj=1 4min x∈BjZBJ−1 J(x−y)dy > 0. Now let us study the uniform boundness with respect to δof the solutions of the limit problems introduced by Theorems 2.1 and 2.2. Proposition 3.2. Let K∈ L(L2(Ω), L2(Ω)) as in (3.16),g:R→Rglobally Lipchitz and consider the map FX:L2(Ω) 7→ L2(Ω) set by FX(u)(x) = fX(x, u(x, t)) = g(mX(x, u)) where the functions fXand mXare given by (2.8) and (2.9). 16 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ On the other hand, we get from Hardy-Littewood maximal inequality a constant ˆ C > 0 such that 1 |Bδ(x)|ZBδ(x) ˜uε(y, s)dy ≤ˆ Ck˜uε(·, t)kL2(Ω) a.e. Ω. Hence, due to (1.4) and (4.25), there exists a constant ˜ C > 0 such that |m(x, ˜uε(x, s))|=|Bδ(x)| |Bδ(x)∩Ωε| 1 |Bδ(x)|ZBδ(x) ˜uε(y, s)dy ≤˜ C for all ∈(0, ε0), s∈[0, T] and x∈Ω proving (4.33). Moreover, we have that (4.34) mΩε(·,˜uε(·, s)) →mX(·, u∗(·, s)) strongly in L2(Ω) as ε→0 for all s∈[0, T]. In fact, for each x∈RNand 0 < ε < ε0, we get from (1.4) and (4.26) that mΩε(x, ˜uε(x, s)) = ZBδ(x) χε(y)dy!−1 ZBδ(x) ˜uε(y, s)dy! → ZBδ(x) X(y)dy!−1 ZBδ(x) u∗(y, s)dy!=mX(x, u∗(x, s)) where mXis defined in (2.9). Hence, since mΩε(·,˜uε) is uniformly bounded in Ω ×[0, T], we can argue as in (4.30) to obtain (4.34) by Convergence Dominated Theorem. Now, using that gis a Lipschitz continuous function, we can get from (4.33) and (4.34) that f(·, uε)* fX(·, u∗) weakly∗in L∞([a, b]; L2(Ω)) where fXis defined in (2.8). Consequently, we can argue as in (4.30) again, to get that Zt 0 e−hε(x)(t−s)f(x, ˜uε(x, s)) ds →Zt 0 e−h0(x)(t−s)fX(x, u∗(x, s)) ds, as ε→0. Consequently, we can pass to the limit in Iε 2getting Iε 2→ZΩ ϕ(x)X(x)Zt 0 e−h0(x)(t−s)fX(x, u∗(x, s)) dsdx, as ε→0. Thus, the limit of the integral equation (4.27) is ZΩ ϕ(x)u∗(x, t)dx =ZΩ ϕ(x)X(x)e−h0(x)tu0(x) + Zt 0 e−h0(x)(t−s)fX(x, u∗(x, s)) dsdx +ZΩ ϕ(x)X(x)Zt 0 e−h0(x)(t−s)ZRN J(x−y)u∗(y, s)dydsdx, ∀ϕ∈L2(Ω), NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 17 which implies u∗(x, t) = e−h0(x)tX(x)u0(x) + Zt 0 e−h0(x)(t−s)X(x)fX(x, u∗(x, s)) ds +Zt 0 e−h0(x)(t−s)X(x)ZRN J(x−y)u∗(y, s)dyds for all t∈[0, T] and a.e. xin Ω. Thus, u∗∈C1([0, T]; L2(Ω)) and satisfies u∗ t(x, t) = X(x)ZRN J(x−y)u∗(y, t)dy −h0(x)u∗(x, t) + X(x)fX(x, u∗(x, t)), u∗(x, t)≡0 for x∈RN\Ω u∗(0, x) = X(x)u0(x) which can be rewritten as (2.11) under assumption (2.10) with Λ ∈L∞(RN) given by Λ(x) = h0(x)− X(x). Finally, let us notice that u∗is unique from Remark 3.4. Indeed, if we re-scale the time t with t=X−1(x)τand set w(x, τ) = u∗(x, X(x)−1τ) we have that wsatisfies equation (3.23) for h(x) = X(x)−1Λ(x)∈L∞(Ω). Thus, u∗is unique which implies that the sequence uεconverges weakly to u∗as ε→0. In this way, we conclude the proofs of Theorems 2.1 and 2.2.  5. A nonlocal equation with local nonlinearity Now, we obtain a nonlocal equation with local nonlinearity from the limit problem given by Theorems 2.1 and 2.2. We consider the limit problem depending on the parameter δ, that is, the equation associated to uδ∈C1([0, T ]; L2(Ω)) which satisfies uδ t(x, t) = X(x)ZRN J(x−y)uδ(y, t)dy −h0(x)uδ(x, t) + X(x)fX(x, uδ(x, t)), uδ(x, t)≡0 for x∈RN\Ω uδ(0, x) = X(x)u0(x). The existence of the family of solutions uδof (3.2) are guaranteed by Proposition 3.2. Also, there exists C > 0, independent of δ > 0, such that, for any bounded interval [a, b]⊂R, sup t∈[a,b] kuδ(·, t)kL2(Ω) ≤C. Hence, if ˜ ·denotes the extension by zero to the whole space RN, we also get that sup t∈[a,b] k˜uδ(·, t)kL2(Ω) ≤C, and then, ˜uδsets a uniformly bounded family in L∞[a, b]; L2(Ω). 18 M. C. PEREIRA AND SILVIA SASTRE-GOMEZ Since L1[a, b]; L2(Ω)is separable, we can extract a subsequence still set by ˜uδsuch that ˜uδ*¯uweakly∗in L∞([a, b]; L2(Ω)), for some ¯u∈L∞[a, b]; L2(Ω). Notice that ¯u(x, t)≡0 in RN\Ω. Then, we can proceed as in Section 4 to prove Theorem 2.3. Since the proof is very similar, we will leave the details to the reader. Here we just pass to the limit in the nonlinear term ZΩ ϕ(x)X(x)fX(x, uδ(·, t)) dx =ZΩ ϕ(x)X(x)g 1 RBδ(x)X(y)dy ZBδ(x) uδ(y, t)dy!dx =ZΩ ϕ(x)X(x)g |Bδ(x)| RBδ(x)X(y)dy 1 |Bδ(x)|ZBδ(x) uδ(y, t)dy!dx. But, it is a direct consequence of Lebesgue Differentiation Theorem and the uniform estimate given by Corollary 3.1. Indeed, from (3.24) and the compact embedding from H1into L2, we have uδ(·, t)→¯u(·, t) strongly in L2(Ω), as δ→0, for any t∈R. Thus, from Lebesgue Differentiation Theorem, we obtain ZΩ ϕ(x)X(x)fX(x, uδ(·, t)) dx →ZΩ ϕ(x)X(x)gX−1(x) ¯u(x, t)dx which leads us to the limit equation (2.15). 6. Appendix In this section we just compute the derivatives of the map Φ : RN7→ Rgiven by Φ(x) = ZB(x) u(y)dy where B(x) = {y∈RN:|x−y|< R}is a ball of radius R > 0 and uis a smooth function. Notice that, due to [31, Lebesgue-Radon-Nikodym Theorem], Φ is absolutely continuous whenever u∈L1(RN). From Taylor’s formula, we have for any v∈RNand t∈Rthat Φ(x+tv)−Φ(x) = ZB(x) {u(y+tv)−u(y)}dy =tZB(x) ∇u(y)·vdy +O(t2). Thus, by Green’s Identity, we get the following expression (6.35) ∇Φ(x)·v=ZB(x) ∇u(y)·vdy =Z∂B(x) u(y)v·N dS where Nis the normal vector on the boundary of the ball ∂B(x). Finally, we observe that this formula (6.35) is in agreement with [17, Theorem 1.11]. NONLOCAL EVOLUTION EQUATIONS IN PERFORATED DOMAINS 19 References [1] W. Allegretto and P. Nistri. On a class of nonlocal problems with applications to mathematical biology, Differential Equations with Applications to Biology, Fields Instit. Commun., Vol. 21 (AMS), pp. 1-14. (1997). [2] F. Andreu-Vaillo, J. M. Maz´on, J. D. Rossi and J. Toledo. Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. AMS, (2010). [3] A. Brillard, D. G´omez, M. Lobo, E. P´erez and T. A. Shaposhnikovad, Boundary homogenization in perforated domains for adsorption problems with an advection term. 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