Robustness of attractors in thin domains for a nonlocal reaction-diffusion equation with sublinear growth term

Discrete and Continuous Dynamical Systems - Series S Vol. 18, No. 7, July 2025, pp. 1906-1922 doi:10.3934/dcdss.2024208 ROBUSTNESS OF ATTRACTORS IN THIN DOMAINS FOR A NONLOCAL REACTION-DIFFUSION EQUATION WITH SUBLINEAR GROWTH TERM Tom´ as Caraballo  ∗1,2, Pedro Mar´ ın-Rubio  1 and Luciano R. N. Rocha  1 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain 2Department of Mathematics, Wenzhou University Wenzhou, Zhejiang Province 325035, China Abstract. This work investigates the dynamics of nonlocal reaction-diffusion equations featuring sublinear growth terms as the domain contracts and undergoes a dimensional reduction. By identifying a limit equation on the lowerdimensional domain, we proceed to compare the associated family of global attractors, revealing the upper-semicontinuity. 1. Introduction. Numerous nonlocal problems have been examined over the past few decades, mainly driven by their practical relevance in real-world applications (e.g. see [9,19,1]). In fact, Further and Grinfeld [8] highlighted that, when dealing with ecology, there is no real justification for assuming that the interactions are local. In particular, considerable focus has been devoted to the nonlocal parabolic PDE ∂u ∂t −a(l(u))∆u=f(1) where lis a nonlocal functional. It is crucial to note that this problem differs significantly from the heat equation, since the Lyapunov structure is lost and it is unclear how to apply manipulations used in the local case. Certainly, distinct conditions may be imposed on the various terms within equation (1) and many of these variations has vastly been studied (e.g. [5,6,7,12,13]). In [3], the authors studied the Dirichlet problem for ∂u ∂t −a(l(u))∆u=f(u) + h(t), where f∈C(R) is a sublinear function, obtaining existence and regularity of pullback attractors in L2and H1 0for the associated dynamical system. Here we are interested in a similar family of problems, but in the autonomous case with Neumann conditions. In addition, we aim to consider these problems in thin domains, 2020 Mathematics Subject Classification. Primary 35B41, 35B30, 37G35; Secondary 35K57, 35B65, 37L99. Key words and phrases. Thin domains, attractors, robustness, nonlocal terms. ∗Corresponding author: Tom´as Caraballo. 1906 NONLOCAL REACTION-DIFFUSION EQUATIONS IN THIN DOMAINS 1907 that is, in a (n+ 1)-dimensional domain Qεshrinking to a ndimensional one in order to investigate how the attractors behave in the shrinking process. Thin domains can be used to improve the understanding of many different fields, e.g. fluid-related processes, such as blood circulation and ocean dynamics. In their seminal work [10], Hale and Raugel introduced a (local) reaction-diffusion equation ∂u ∂t −∆u+αu =−f(u)−G with Neumann conditions within a (n+ 1)-dimensional thin domain Qε,n= 1,2, which is bounded above by the graph of a C1function with certain conditions. Among their results is the fact that the attractors Aεultimately converge to a distinctive n-dimensional attractor Aassociated with a limit equation. Additionally, they establish upper semicontinuity in the limit. These results are strongly relied in the gradient structure of the attractors and on computations that only can be made in the local problem. Arrieta and Santamar´ıa [2] improve the results allowing the domains to have greater dimensions and obtaining better rates for the convergence of the attractors. In the nonlocal context, Pereira and Rossi [14] work on the problem with a nonlocal effect of the form ZΩ1×Ω2 Jε(x−y)(uε(y)−uε(x))dy, where Jε(z) = J(z1, εz2) and Ω1×Ω2⊂RN1×RN2. They obtained that, as εgoes to 0, the solutions uεconverge to some u0whose mean with respect to Ω2satisfies an specific equation. For more on thin domains, we may refer to [15] and [16] and the references therein. In this paper, we study existence, regularity, convergence and upper-semicontinuity of global attractors Aε,ε∈(0,1], for the solutions of ∂v ∂t −a(ℓε(v))∆v+αv =f(v) + G in the thin domain Qε×(0,+∞), where Qε= Ω ×(0, ε) is a bounded open set of Rn×Rand f∈ C(R) is a sublinear function. Further conditions will be given below. We will proceed with a simple thin domain due to the restrictive nature of the nonlocal term, which obstruct the application of our technique under the broader hypotheses associated with a more generalized domain varying along the x-direction. The problem is fulfilled with initial and Neumann boundary conditions. The structure of the paper is as follows. In Section 2 the problem is properly stated and some results concerning existence and regularity of solutions are proved. In Section 3 the existence of global attractors is ensured. Finally, in Section 4 we establish the upper-semicontinuity for the attractors toward the attractor of a suitable dynamical system generated by a limit problem when the parameter goes to zero. 2. Statement, existence and regularity. Let Q= Ω×(0,1) be a bounded open set of Rn×Rsuch that ∂Ω is sufficiently regular. For ε > 0, consider the squeezing operator Φε:Rn×R→Rn×R (x, y)7→ (x, εy). 1908 TOM´ AS CARABALLO, PEDRO MAR´ IN-RUBIO AND LUCIANO R. N. ROCHA We denote by Qεthe image of Qthrough Φε. In Qε, consider the following problem for a nonlocal sublinear parabolic equation:      ∂v ∂t −a(ℓε(v))∆v+αv =f(v) + Gin Qε×(0,+∞) ∂νεv= 0 on ∂Qε×(0,+∞), v((x, y),0) = u0(x, y) in Qε, (2) where α > 0, a∈C(R;R+) is a locally Lipschitz function which is bounded below by a positive constant m, that is, 0< m ≤a(s)∀s∈R.(3) For the nonlocal term, we assume that the functionals ℓεare such that they can be written ℓε(v) = 1 |Qε|RQεgε(X, Y )v(X, Y )dXdY , where gε∈L2(Q) converge strongly in L2(Q) to some g0which does not depends on the Y direction, that is, g0(X, Y ) = g0(X). Additionally, let G∈C(˜ Q), where ˜ Q= Ω ×(0, ε0) for some ε0∈(0,1). Finally, for the sublinear term, we assume that f∈C(R) and that there exist constants η > 0 and Cf>0 such that |f(v)| ≤ Cf(1 + |v|)∀v∈R,(4) (f(v)−f(w))(v−w)≤η(s−r)2∀v, w ∈R(5) and, for some 0 < µ < α there exists cµ>0 such that f(v)v≤µv2+cµ∀v∈R.(6) Now we perform a change of variable to obtain an equivalent problem. Consider the operator Ψε:L2(Qε)→L2(Q) v7→ v◦Φε. Then vε∈L2(Qε) is a solution to (2) if, and only if, uε:= Ψε(vε)∈L2(Q) is a solution to      ∂u ∂t −a(lε(u))(∆xu+1 ε2∆yu) + αu =f(u) + Gεin Q×(0,+∞) ∂νxu+1 ε2∂νyu= 0 on ∂Q ×(0,+∞), u((x, y),0) = uε 0:= u0(x, εy) in Q, (7) where Gε(x, y) = G(x, εy) and lε∈(L2(Q))′is given by lε(u) = 1 |Ω|ZQ gε(x, εy)u(x, y)dxdy. Here, we employed the relation ℓε(v) = lε(u). It is more convenient to work on a fixed domain, so from now on, we only deal with (7). We will compare the solutions of (7) with the solutions of the problem      ∂u ∂t −a(˜ l(u))∆xu+αu =f(u) + G0in Ω ×(0,+∞) ∂νxu= 0 on ∂Ω×(0,+∞), u(x, 0) = (Mu0)(x) := R1 0u0(x, y)dy in Ω, (8) where G0(x) := G(x, 0) and ˜ l(u) = 1 |Ω|RΩg0(x)u(x)dx. For short, we denote the operators ∇εu:= (∇xu, 1 ε∇yu), ∆εu:= ∆xu+1 ε2∆yu,Aεu:= −∆εu+Iand Au := (−∆ + I)u. NONLOCAL REACTION-DIFFUSION EQUATIONS IN THIN DOMAINS 1909 On H1(Q) consider the inner product given by ((u, v))ε:= ZQ∇xu∇xv+1 ε2∇yu∇yv+uvdxdy and ∥·∥εthe generated norm. We will denote the normed space (H1(Q),∥·∥ε) by H1 ε. Remark 1. The analogous problem with Dirichlet boundary conditions can also be analyzed. In this case, for εsmall enough, the problem has a unique solution uε∈C([0, T]; H1 0) (see [3]) and it is not difficult to show that there exists a function u0∈C([0, T]; H1 0) such that ∇yu0= 0 and uε→u0as ε→0 for every T > 0. The Poincar´e Inequality (see [18]) implies (the less interesting consequence) that u0is the null function. Definition 1. Given uε 0∈L2(Q), a weak solution to (7) is a function uεthat belongs to L2(0, T;H1(Q)) ∩L∞(0, T;L2(Q)) for all T > 0, with uε(0) = uε 0, such that d dt(uε(t), v) + a(lε(uε(t)))(∇εuε(t),∇εv) + α(uε(t), v) =(f(uε(t)), v)+(Gε, v),∀v∈H1(Q).(9) If uεis a weak solution of (7), then equation (9), condition (4) and the continuity of aimply that duε dt ∈L2(0, T; (H1(Q))′). By [4, Chapter 2, Theorem 1.8], we have Proposition 1. If uεis a weak solution to (7), then uε∈C([0,∞); L2(Q)) and the following energy equality holds 1 2∥uε(t)∥2 L2(Q)+Zt sa(lε(uε(r)))(∥∇xuε(r)∥2 L2(Q)+1 ε2∥∇yuε∥2 L2(Q))+α∥uε∥2 L2(Q)dr =1 2∥uε(s)∥2 L2(Q)+Zt s (f(uε(r)), uε(r))dr +Zt s (Gε, uε(r))dr ∀0≤s≤t. (10) In particular, the initial data uε(0) = uε 0in Definition 1makes sense. It is also convenient to introduce the notion of a more regular solution. Definition 2. A strong solution to (7) is a weak solution uεthat also satisfies uε∈L2(0, T;D(A)) ∩L∞(0, T;H1(Q)) for all T > 0. Again, [4, Chapter 2, Theorem 1.8] can be used to prove the following result. Proposition 2. If uεis a strong solution to (7), then u∈C([0,+∞); H1(Ω)) and the following energy equality holds for all 0≤s≤t: 1 2∥∇εuε(t)∥2 L2(Q)+Zt sa(lε(uε(r)))∥ − ∆εuε(r)∥2 L2(Q)+α∥∇εuε(r)∥2 L2(Q)dr =1 2∥∇εuε(s)∥2 L2(Q)+Zt s (f(uε(r)) + Gε,−∆εuε(r))dr ∀0≤s≤t. Theorem 1. Assume that a∈Liploc(R)satisfies (3),f∈C(R)fulfils (4),(5) and (6)and that, for every 0< ε < 1, the functional lε∈(L2(Q))′is given by lε(u) = 1 |Ω|RQgε(x, εy)u(x, y)dxdy, where gε∈L2(Q)converges to g0strongly in L2(Q),and that g0does not depend on the ydirection, that is, g0(x, y) = g0(x). Assume that G∈C(˜ Q), where ˜ Q= (0, ε0)×Ωfor some ε0∈(0,1). Then, for each 0< ε < ε0and uε 0∈L2(Q), problem (7)has a unique weak solution, denoted 1910 TOM´ AS CARABALLO, PEDRO MAR´ IN-RUBIO AND LUCIANO R. N. ROCHA by uε(·;uε 0). Moreover, this solution behaves continuously in L2(Q)with respect to initial data. In addition, for every δ > 0and T > δ, it satisfies that uε∈ C((δ, T ]; H1(Q)) ∩L2(δ, T;D(Aε)). If uε 0∈H1(Q), then uεis a strong solution. Proof. Fix ε∈(0, ε0). Uniqueness of solution and continuity with respect to initial data. Let 0< ε < ε0and suppose that uε 1and uε 2are two weak solutions to (7) corresponding to initial values uε 1,0, uε 2,0∈L2(Q) respectively. From the energy equality, 1 2 d dt∥uε 1(t)−uε 2(t)∥2 L2(Q)+a(lε(uε 1))(∇ε[uε 1−uε 2],∇ε[uε 1−uε 2]) + α∥uε 1−uε 2∥2 L2(Q) =[a(lε(uε 2)) −a(lε(uε 1))](∇εuε 2,∇ε[uε 1−uε 2]) + (f(uε 1)−f(uε 2), uε 1−uε 2) a.e. t∈(0, T). Let Mlbe a uniform upper bound for ∥lε∥(L2(Q))′when ε∈(0, ε0) and Lathe Lipschitz constant for ain the compact set [−Mlmax{∥uε i(t)∥L∞(0,T ;L2(Q)) :i= 1,2}, Mlmax{∥uε i∥L∞(0,T ;L2(Q)) :i= 1,2}]. Then (3) and (5) imply 1 2 d dt∥uε 1(t)−uε 2(t)∥2 L2(Q)+ min{m, α}∥uε 1(t)−uε 2(t)∥2 ε ≤La Ml |Ω|∥uε 2∥ε∥uε 1(t)−uε 2(t)∥ε∥uε 1(t)−uε 2(t)∥L2(Q)+η∥uε 1(t)−uε 2(t)∥2 L2(Q). Thanks to the Young inequality, d dt∥uε 1(t)−uε 2(t)∥2 L2(Q)≤(L2 aM2 l 2|Ω|2min{m, α}∥uε 2(t)∥2 ε+ 2η)∥uε 1(t)−uε 2(t)∥2 L2(Q). Therefore, ∥uε 1(t)−uε 2(t)∥2 L2(Q)≤exphZt 0 L2 aM2 l 2|Ω|2min{m, α}∥u2(s)∥2 ε+2η dsi∥uε 1,0−uε 2,0∥2 L2(Q). From this, we obtain uniqueness (setting uε 1,0=uε 2,0) and continuity with respect to initial data. Existence of weak solution. Consider ε∈(0, ε0), T > 0 fixed and {wε j} ⊂ H1(Q) a Hilbert basis of L2(Q) formed by eigenfunctions of Aεwith Neumann boundary condition in Q, with corresponding eigenvalues {λε 1, λε 2, . . . }for every integer n≥1. Denote by uε n(·;uε 0) = Pn j=1 φε nj(·)wε j,uε n(·) for short, the Galerkin approximation solution (cf. [17]) in some interval (0, tn) (with tn≤T) of d dt(uε n(t), wε j)−a(lε(uε n(t)))(∆εuε n(t), wε j) + α(uε n(t), wε j) =(f(uε n(t)), wε j)+(Gε, wε j), (uε n(0), wε j)=(uε 0, wε j), j = 1, . . . , n. (11) Multiplying by φε nj(t), summing from j= 1 to nand using (3), we obtain d dt∥uε n(t)∥2 L2(Q)+ 2m∥∇εuε n(t)∥2 L2(Q)+ 2α∥uε n(t)∥2 L2(Q) ≤2(f(uε n(t)), uε n(t)) + 2(Gε, uε n(t)) a.e. t∈(0, tn). NONLOCAL REACTION-DIFFUSION EQUATIONS IN THIN DOMAINS 1911 By the Young inequality and (3), we have d dt∥uε n(t)∥2 L2(Q)+ min{m, α}∥uε n(t)∥2 ε ≤2C2 f|Ω| α+ 2Cf∥uε n(t)∥2 L2(Q)+2 α∥Gε∥2 L2(Q)a.e. t∈(0, tn) and min{m, α}ZT 0 ∥uε n(s)∥2 εds ≤∥uε 0∥2 L2(Q)+4C2 f|Ω|+ 2∥G∥2 L2(Q) αT+4C2 f αZT 0 ∥uε n(s)∥2 L2(Q)ds. (12) Hence, from the Gronwall lemma we deduce that {uε n}nis well-defined on the whole interval [0, T] and bounded in L∞(0, T;L2(Q)) ∩L2(0, T ;H1(Q)). Taking into account that each uε n∈C([0, T]; L2(Q)), we obtain that there exists a constant Csuch that ∥uε n(t)∥L2(Q)≤Cfor every t∈[0, T ] and n≥1. Since ais continuous and bounded in the compact set [−CMl, CMl], so there exists a constant Msuch that a(lε(uε n(t))) ≤Mfor every t∈[0, T ] and n≥1. So, ZT 0|a(lε(uε n))|2∥∆εuε n(t)∥2 (H1(Q))′+α2∥uε n(t)∥2 L2(Q)dt ≤max{M2, α2}ZT 0 ∥uε n∥2 εdt. This implies that {−a(lε(uε n))∆εuε n+αuε n}nis bounded in L2(0, T; (H1(Q))′). Using (4), it follows that {f(uε n)}nis bounded in L2(0, T;L2(Q)). Then, {d dt uε n}n is bounded in L2(0, T, (H1(Q))′). Since H1(Q)⊂⊂ L2(Q)⊂(H1(Q))′, the Aubin-Lions Lemma guarantees that there exist a subsequence of {uε n}n(relabeled the same) and uε∈C([0, T]; L2(Q))∩ L2(0, T;H1(Q)) such that uε n(t)→uε(t) strongly in L2(Q) a.e. t∈(0, T ). Now, it is not difficult to conclude that uεin fact satisfies equation (7). Actually, from the continuity of f, since f(uε n(x, t)) converge to f(uε(x, t)) a.e. in Q×(0, T ), from the boundedness of {f(uε n)}nand [11, Lemma 1.3], it follows that f(uε n)⇀ f(uε) in L2(0, T;L2(Q)). Let us prove that, given w∈H1(Q) and φ∈D(0, T ), it holds (a(lε(uε n))∇εuε n,∇εwφ)→(a(lε(u))∇εuε,∇εwφ). Indeed, observe that lε∈L2(Q) and a∈C(R), along with the strong convergence of {uε n}nin L2(Q), imply that a(lε(uε n(t)))∇εwφ →a(lε(uε(t)))∇εwφ a.e. in Q× (0, T). The Dominated Convergence Theorem implies that this convergence also holds on L2(Q×(0, T)). Since ∇εuε n⇀∇εuε∈L2(Q×(0, T)), it follows that ZT 0ZQ a(lε(uε n))∇εuε n∇εwφdxdyds →ZT 0ZQ a(lε(u))∇εuε∇εwφdxdyds. Then, one may pass to the limit in (11) and since ∪n∈Nspan{wε 1, . . . , wε n}is dense in H1(Q), we conclude that uis a weak solution on (0, T). Finally, since the fixed value T > 0 was arbitrary, we may concatenate solutions and, thanks to the uniqueness, obtain the (global) weak solution u. 1912 TOM´ AS CARABALLO, PEDRO MAR´ IN-RUBIO AND LUCIANO R. N. ROCHA Regularizing effect. Multiplying by (λε j−1)φε nj(t) in (11), summing from j= 1 to nand using (3) and the Young inequality, we obtain d dt∥∇εuε n(t)∥2 L2(Q)+m∥∆εuε n(t)∥2+α∥∇εuε n(t)∥2 L2(Q) ≤4C2 f|Ω| m+4C2 f m∥uε n(t)∥2 L2(Q)+2 m∥Gε∥2 L2(Q).(13) In particular, integrating between sand tone can deduce that ∥∇εuε n(t)∥2 L2(Q)+mZt s ∥∆εuε n(t)∥2 L2(Q)dr ≤∥∇εuε n(s)∥2 L2(Q)+4C2 f|Ω|+ 2∥Gε∥2 L2(Q) m(t−s) + 4C2 f mZt s ∥uε n(r)∥2 L2(Q)dr. (14) Integrating with respect to sbetween 0 and t, follows that t∥∇εuε n(t)∥2 L2(Q)≤4C2 f|Ω|+ 2∥Gε∥2 L2(Q) mT2+4C2 fT m+ 1ZT 0 ∥uε n(r)∥2 εdr. Therefore, for all t∈[δ, T ] with δ∈(0, T ), it holds ∥∇εuε n(t)∥2 L2(Q)≤4C2 f|Ω|+ 2∥Gε∥2 L2(Q) δm T2+4C2 fT+m mδ ZT 0 ∥uε n(r)∥2 εdr From this, (12) and the existence of weak solution, the sequence {uε n}nis bounded in L∞(δ, T ;H1(Q)). On the other hand, taking s=δand t=Tin (14), and using the previous estimates, it arises that {uε n}nis bounded in L2(δ, T ;D(Aε)). As a consequence, {(uε n)′}is bounded in L2(δ, T ;L2(Q)), and thanks to the uniqueness of the weak solution, it holds that {uε n}nconverges to uweakly in L2(δ, T ;D(Aε)) and {d dt uε n}nconverges to d dt uεweakly in L2(δ, T;L2(Q)). Then we have uε∈ L2(δ, T ;D(Aε)) ∩C([0, T ]; H1(Q)). Strong solution. Assume uε 0∈H1(Q). We shall prove that the corresponding solution satisfies uε∈L2(0, T;D(A)) ∩L∞(0, T;H1(Q)). Indeed, in this case we can integrate (13) between 0 and t∈[0, T] and get inequality (14) with s= 0. Since {uε n}nis bounded in L∞(0, T;L2(Q)), then it is bounded in L∞(0, T ;H1(Q)) ∩L2(0, T;D(Aε)) too. Since the weak solutions are unique, we have uε n ∗ ⇀ u in L∞(0, T;H1(Q)) and uε n⇀ u in L2(0, T;D(Aε)). and uis a strong solution. With completely analogous arguments we obtain Proposition 3. Assume that the function ais locally Lipschitz and satisfies (3), f∈C(R)fulfils (4),(5)and (6)and ˜ l∈(L2(Ω))′is given by ˜ l(u) = 1 |Ω|RΩg0(x)u(x) dx, where g0∈L2(Ω). Assume also that G∈C(˜ Q). Then, for any u0∈L2(Ω), the problem (8)possesses a unique solution, denoted by u(·) = u(·;u0).Moreover, this solution behaves continuously in L2(Ω) with respect to initial data. In addition, for every δ > 0and T > δ, it satisfies that u∈C((δ, T ]; H1(Ω)) ∩L2(δ, T;D(A)). In fact, if the initial condition u0∈H1(Ω), then u∈C([0, T ]; H1(Ω))∩L2(0, T ;D(A)) for every T > 0, i.e. uis a strong solution. NONLOCAL REACTION-DIFFUSION EQUATIONS IN THIN DOMAINS 1913 3. Global attractors. Here after that the assumptions of Theorem 1and Proposition 3hold and consider ε∈(0, ε0). Let {Sε(t)}t≥0be the semigroup generated by (7). These semigroups can be defined in both H1(Q) or L2(Q). Since no confusion arises, we will use the same notation for these semigroups without any more indication of restriction. Analogously {S0(t)}t≥0will denote the semigroup generated by (8) in H1(Ω) or L2(Ω) (although for our purposes we will extend it trivially also in the y-direction in a constant way to have it defined in Q). Remark 2. Assumption G∈C(˜ Q) introduced in the previous two results implies that one can apply the Lebesgue Dominated Convergence Theorem to Gε(x, y) := G(x, εy), namely ∥Gε∥2 L2(Q)=ZQ G(x, εy)2dxdy →ZQ G(x, 0)2dxdy as ε→0. Therefore, for εsmall enough (but for simplicity we denote with the same ε0) it holds that ∥Gε∥2 L2(Q)≤ ∥G0∥2 L2(Q)+ 1 for all ε∈(0, ε0).This is important in order to gain uniform estimates (independent of ε) in what follows. Lemma 1. The dynamical system (L2(Q),{Sε(t)}t≥0)associated to problem (7) has an absorbing set in L2(Q)which does not depends on ε, that is, there exist a constant ρ1>0such that for every bounded set B⊂L2(Q), there exists a time t0(B)such that ∥uε(t;u0)∥L2(Q)≤ρ1for any u0∈B,t > t0(B)and 0< ε ≤ε0. Proof. Consider B⊂L2(Q) bounded, 0 < ε < ε0and u0∈B. The energy equality (10) for the solution uεto (7), assumption (3) and the Cauchy-Schwartz inequality give 1 2 d dt∥uε(t)∥2 L2(Q)+m∥∇εuε(t)∥2 L2(Q)+α∥uε(t)∥2 L2(Q) ≤∥Gε∥L2(Q)∥uε(t)∥L2(Q)+ZQ f(uε(t))uε(t)dxdy a.e. t > 0. Then, by (6) and the Young Inequality, it holds d dt∥uε(t)∥2 L2(Q)+ 2m∥∇εuε(t)∥2 L2(Q)+ (α−µ)∥uε(t)∥2 L2(Q) ≤∥Gε∥2 L2(Q) (α−µ)+ 2cµ|Ω|(15) In particular, from the Gronwall Lemma one deduces that ∥uε(t)∥2 L2(Q)≤ ∥u0∥2 L2(Q)e−(α−µ)t+∥Gε∥2 L2(Q)+ 2(α−µ)cµ|Ω| (α−µ)2∀t≥0. After Remark 2, from above we have that there is t0(B)>0 such that ∥uε(t)∥2 L2(Q)≤ρ2 1:= 2∥G0∥2 L2(Q)+ 1 + 2(α−µ)cµ|Ω| (α−µ)2∀t≥t0(B). Remark 3. If Cf< α, condition (4) suffices to obtain the result and (6) can be suppressed. For the next result, we will make use of the following lemma (e.g., cf. [17]). 1914 TOM´ AS CARABALLO, PEDRO MAR´ IN-RUBIO AND LUCIANO R. N. ROCHA Lemma 2. Let Xand Ybe Banach spaces such that Xis reflexive and continuously embedded in Y. Assume that {un}is a bounded sequence in L∞(0, T;X)such that un⇀ u in Lq(0, T ;X)for some q∈[1,∞)and u∈C([0, T ]; Y). Then, u(t)∈X and ∥u(t)∥X≤lim inf n→∞ ∥un∥L∞(0,T ;X),∀t∈[0, T]. Lemma 3. Under the assumptions of Theorem 1, for ε∈(0, ε0)the dynamical system (L2(Q),{Sε(t)}t≥0)has an absorbing set in H1 ε(Q)which does not depend on ε, that is, there exists a constant ρ2>0such that for every bounded set B⊂L2(Q), there is a time t1(B)such that ∥uε(t;u0)∥ε≤ρ2for any u0∈B, t > t1(B)and 0< ε ≤ε0. Proof. Analogously to the proof of Lemma 1, we start with the energy equality, particularly from estimate (15), which implies that d dt∥uε(t)∥2 L2(Q)+c∥uε(t)∥2 ε≤∥Gε∥2 L2(Q) α−µ+ 2cµ|Ω|a.e.t > 0 where c= min{2m, α −µ}. It follows from this and Lemma 1that cZt+1 t ∥uε(s)∥2 εds ≤∥Gε∥2 L2(Q) (α−µ)+ 2cµ|Ω|+∥uε(t)∥2 L2(Q) ≤∥Gε∥2 L2(Q) (α−µ)+ 2cµ|Ω|+ρ2 1∀t>t0(B).(16) Actually observe that this inequality also holds for the Galerkin approximations used in the proof of Theorem 1. Multiplying (11) by (λε j−1)φε nj(t), summing from j= 1 to nand using (3), (4) and the Young inequality, we obtain d dt∥∇εuε n(t)∥2 L2(Q)+ 2m∥∆εuε n(t)∥2 L2(Q)+ 2α∥∇εuε n(t)∥2 L2(Q) ≤4C2 f|Ω| m+4C2 f m∥uε n(t)∥2 L2(Q)+2∥Gε∥2 L2(Q) m+m∥∆εuε n(t)∥2 L2(Q)a.e.t > 0.(17) In particular, tidying the expression and neglecting a positive term in the left hand side, d dt∥∇εuε n(t)∥2 L2(Q)≤4C2 f|Ω| m+4C2 f m∥uε n(t)∥2 L2(Q)+2∥Gε∥2 L2(Q) ma.e.t > 0. Now assume that t0(B)< t −1≤s<t. By integrating the above on [s, t] it yields ∥∇εuε n(t)∥2 L2(Q)≤4C2 f|Ω| m+4C2 f mρ2 1+2∥Gε∥2 L2(Q) m+∥∇εuε n(s)∥2 L2(Q). Integrating again between t−1 and twith respect to s, it holds ∥∇εuε n(t)∥2 L2(Q)≤Zt t−1 ∥∇εuε n(s)∥2 L2(Q)ds +4C2 f|Ω| m+4C2 f mρ2 1+2∥Gε∥2 L2(Q) m. Now it follows from (16) that ∥∇εuε n(t)∥2 L2(Q)≤∥Gε∥2 L2(Q) c(α−µ)+2 ccµ|Ω|+ρ2 1 c+4C2 f|Ω| m+4C2 f mρ2 1+2∥Gε∥2 L2(Q) m for all t>t0(|u0|)+1. NONLOCAL REACTION-DIFFUSION EQUATIONS IN THIN DOMAINS 1921 Taking into account (5) and the estimate (27), we obtain 1 2 d dt∥∇εt(uε−u)(t)∥2 L2(Q)+m 2∥∆εt(uε−u)(t)∥2 L2(Q)+α∥∇εt(uε−u)(t)∥2 L2(Q) ≤1 m(η2∥t(uε−u)(t)∥2 L2(Q)+∥t2(Gε−G0)∥2 L2(Q)+2L2 a |Ω|2(∥gε(x, y)−g0(x)∥2 L2(Q) × ∥uε(x, y)∥2 L2(Q)+∥g0∥2 L2(Ω)∥uε−u∥2 L2(Q))∥∆εtu∥2 L2(Q)+∥uε−u∥2 L2(Q)) Using Theorem 2, we find a constant C=C(R) such that d dt∥∇εt(uε−u)(t)∥2 L2(Q)+m∥∆εt(uε−u)(t)∥2 L2(Q)+α∥∇εt(uε−u)(t)∥2 L2(Q) ≤Cd(ε)(eCt +eCtt2∥∆εu∥2 L2(Q)+∥∆εu∥2 L2(Q)t2) Since uis a strong solution, we have ∆εu∈L2(0, T ;L2(Q)). So, integrating above from 0 to t, we obtain (with a redefined constant) t2∥∇ε(uε−u)(t)∥2 L2(Q)+Zt 0 m∥∆εs(uε−u)(s)∥2 L2(Q)+α∥∇εs(uε−u)(s)∥2 L2(Q)ds ≤d(ε)CeCt. This inequality and Theorem 2give us the result. Theorem 4. The attractors Aεare upper-semicontinuous at ε= 0, that is, for any r > 0, there exists a ε1>0such that Aε⊂ NH1 ε(A0, r)∀ε∈(0, ε1). Proof. Fix t0>0 and let A0be the attractor of S0(t). Then, given r > 0, there exists a time τ0> t0such that, for any t≥τ0, inf ψ0∈A0 ∥S0(t)v−ψ0∥H1(Ω) ≤r 2 for any v∈H1(Ω) with ∥v∥H1(Ω) ≤ρ2. If φε∈H1(Q) and ∥φε∥ε≤ρ2, we have, thanks to Theorem 3, that for t0≤t≤τ0, ∥Sε(t)φε−S0(t)(Mφε)∥ε≤d(ε)1 2 t0 (˜ K1(ρ2))1 2e ˜ K1(ρ2)τ0 2. Applying the triangle inequality above, we infer that ∥Sε(τ)φε−ψ0∥ε≤r 2+d(ε)1 2 t0 (˜ K1(ρ2))1 2e ˜ K1(ρ2)τ0 2.(28) In particular, taking into account Lemma 7, this inequality holds for any φε∈ Aε. Since Sε(τ0)Aε=Aε, (28) implies the stated result, where ε1< ε0is a positive number satisfying ε1 t0 (˜ K1(ρ2))1 2e ˜ K1(ρ2)τ0 2≤r 2. Acknowledgments. Partially supported by the projects P18-FR-4509 (Junta de Andaluc´ıa/FEDER/US) and PGC2018-096540-B-I00 and PID2021-122991NB-C21 (MCIU-AEI-Spain/FEDER). 1922 TOM´ AS CARABALLO, PEDRO MAR´ IN-RUBIO AND LUCIANO R. N. ROCHA REFERENCES [1] M. Anguiano, P. E. Kloeden and T. Lorenz, Asymptotic behaviour of nonlocal reactiondiffusion equations,Nonlinear Anal.,73 (2010), 3044-3057. [2] J. M. Arrieta and E. Santamar´ıa, Distance of attractors of reaction-diffusion equations in thin domains,J. Differential Equations,263 (2017), 5459-5506. [3] T. Caraballo, M. Herrera-Cobos and P. Mar´ın-Rubio, Long-time behavior of a nonautonomous parabolic equation with nonlocal diffusion and sublinear terms,Nonlinear Anal., 121 (2015), 3-18. [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [5] M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems,Positivity, 3(1999), 65-81. [6] M. Chipot and M. Siegwart, On the asymptotic behaviour of some nonlocal mixed boundary value problems, Nonlinear Analysis and Applications: To V. Lakshmikantam on his 80th Birthday, Kluwer Acad. Publ, Dordrecht, (2003), 431-449. [7] N. H. Chang and M. Chipot, On some model diffusion problems with a nonlocal lower order term,Chinese Ann. Math. Ser. B,24 (2003), 147-166. [8] J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics,J. Math. Biology,27 (1989), 65-80. [9] J. Garc´ıa-Meli´an and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal.,8(2009), 2037-2053. [10] J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. [11] J. L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Paris, 1969. [12] S. de Menezes, Remarks on weak solutions for a nonlocal parabolic problem,Int. J. Math. Math. Sci.,2006 (2006), 82654, 10 pp. [13] A. Ovono, Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations,134 (2019), 1-16. [14] M. C. Pereira and J. D. Rossi, Nonlocal problems in thin domains,J. Differential Equations, 263 (2017), 1725-1754. [15] M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations,J. Differential Equations,173 (2001), 271-320. [16] G. Raugel, Dynamics of Partial Differential Equations on Thin Domains, Lecture Notes in Math., 1609 Springer-Verlag, Berlin, 1995 [17] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. [18] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr., 49. American Mathematical Society, Providence, RI, 1997. [19] Z. Szymanska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interaction,Math. Models Methods Appl. Sci.,19 (2009), 257-281. Received July 2024; revised November 2024; early access November 2024.