Robustness of time-dependent attractors in H1-norm for nonlocal problems

DISCRETE AND CONTINUOUS doi: DYNAMICAL SYSTEMS SERIES B Volume , Number 0, pp. 000–000 ROBUSTNESS OF TIME-DEPENDENT ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS Tom´ as Caraballo, Marta Herrera-Cobos & Pedro Mar´ ın-Rubio Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, SPAIN To Professor Igor Chueshov, in Memoriam Abstract. In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35–50]. 1. Introduction and existence results. Many nonlocal problems have been analysed in the last few decades due to their usefulness in real applications (e.g. cf. [23,4,25,40,3]). Namely, many authors have been interested in studying the nonlocal parabolic equation ∂u ∂t −a(l(u))∆u=f, where ais a continuous function and l∈(L2(Ω))0, i.e. l(u) = lg(u) = ZΩ g(x)u(x)dx. From a biological point of view, the function umight represent the density of a population. Additional assumptions could be imposed on the function ato better reflect the behaviour of the community. For instance, to model species with a tendency to leave crowded zones, a natural assumption would be to assume that a is an increasing function of its argument. On the other hand, if we are dealing with species attracted by growing population, one would assume ato decrease. This equation has been used in epidemic theory and from a physical point of view, to study the heat propagation (for more details cf. [13,14]). It is worth highlighting that the above equation is not a trivial perturbation of the heat equation and serious difficulties arise in different contexts. For instance, the existence of a Lyapunov function is not guaranteed in a general framework. Additional requirements (see [14] for more details) or more specific nonlocal operators, 2000 Mathematics Subject Classification. Primary: 35B41, 35B65, 35K57, 35Q92, 37L30. Key words and phrases. Non-autonomous nonlocal reaction-diffusion equations; pullback attractors; regularity of attractors; upper semicontinuous behaviour. Partially funded by the projects MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM- 1492 (Junta de Andaluc´ıa). 1 2 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO which are strongly related to the diffusion terms (see [17,15]), are needed to build this structure. Many authors have been interested in studying the asymptotic behaviour of the solutions to problems of the same kind as the one presented above. For instance, in [13] Chipot and Lovat establish a comparison result between two stationary solutions and the solution of the evolution problem. Then, using this result, they prove the convergence of the solution of the evolution problem towards a stationary solution. Later, Chipot and Molinet [14] generalise the results obtained in [13] dealing with a continuum of steady states, making use of dynamical systems. In [16] Chipot and Siegwart consider a more general elliptic operator to study the long-time behaviour of the solutions. Chang and Chipot [11] analyse the same kind of results as in [13], however, in this case the authors deal with two nonlocal operators. Another interesting study is the one by Chipot and Zheng in [18], where the authors analyse the convergence of the solution of the evolution problem to one of the equilibria without assuming uniqueness of stationary solutions. Among several other results, Andami Ovono [1] analyses the existence of the compact global attractor in L2(Ω). If falso depends on the unknown u, the study of the stationary solutions is not trivial at all, and the natural generalization for the analysis of the long-time behaviour of the solutions is to consider the theory of attractors. Although an autonomous approach is almost new in this setting, it might be meaningful (and more general) to consider time-dependent terms in the model and then there are several approaches from the point of view of non-autonomous dynamical systems, like skew-product flows (see Sell [38]), uniform attractors and their kernel sections (cf. Chepyzhov and Vishik [12]) and pullback attractors (see Kloeden and Schmalfuß [30,31] and Kloeden [28], also related to random dynamical systems [21]). This last approach allows us to minimize the assumptions on the forcing terms and the resultant objects are strictly invariant in a suitable non-autonomous-dynamical- system sense, unlike what happens with uniform attractors. Furthermore, pullback attractors let us study the behaviour of the current system considering initial times that come from the past (e.g. cf. [29] for more details). Many new results have appeared over the last years related to pullback attractors. Some authors have been interested in studying the pullback attractor in the classical sense, i.e. the pullback attractor of solutions starting in fixed bounded sets. Others, though, have employed the concept of attraction related to a class of families, called universe D, which is made up of sets which are allowed to move in time and are usually defined in terms of a tempered condition (e.g. cf. [19,9,10]). This last approach has been used recently to study nonlocal problems (cf. [5,7]). For instance, in [5] the existence of pullback attractors in L2(Ω) and H1 0(Ω) is established for a nonautonomous parabolic equation with nonlocal diffusion and sublinear terms. Later in [7], continuing in a single-valued framework (the nonlocal viscosity is locally Lipschitz), the existence of these families is analysed for a non-autonomous nonlocal reaction-diffusion equation. In addition, in [6] the existence of pullback attractors in L2(Ω) is analysed in a multi-valued framework for the non-autonomous nonlocal reaction-diffusion problem      ∂u ∂t −(1 −ε)a(l(u))∆u=f(u) + εh(t) in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x) in Ω, ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 3 where ε∈[0,1] and no locally Lipschitz assumption on the function aor monotonicity on the nonlinearity fare imposed (so lack of uniqueness). Moreover, the upper semicontinuous behaviour of attractors in L2-norm is studied. Namely, it is proved that the family of pullback attractors converges to the compact global attractor associated to the autonomous problem when εgoes to 0. [Do not confuse this notion, the upper-semicontinuous behaviour of attractors, with upper-semicontinuous process (see Definition 4). While being an upper-semicontinuous multi-valued process is a property only related with a two-parameter semigroup, the upper-semicontinuous behaviour of attractors takes into account all the processes indexed by the parameter in which it is taken the limit and it analyses an asymptotic property of a whole family of problems.] In this paper we improve the results given in [6], analysing existence of pullback attractors in H1 0(Ω) as well as their upper semicontinuous behaviour in H1-norm. According to [6, Section 6] we consider the parameterized family (ε∈[0,1]) of non-autonomous nonlocal reaction-diffusion problems (Pε)     ∂u ∂t −g1(ε)a(l(u))∆u= ˜g1(ε)f(u) + g0(ε)h(t) in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x) in Ω, where Ω is a bounded open subset of RNof class C1,1,τ∈R,the nonlocal diffusion term fulfils that there exists m > 0 such that a∈C(R; [m, ∞)),(1) l∈(L2(Ω))0.(2) Concerning the family of perturbed coefficients, varying with the parameter ε, suppose that g1∈C([0,1]; (0,∞)),˜g1∈C([0,1]; [0,∞)), g0∈C([0,1]).(3) Actually, at some stages of the paper –not from the very beginning– it will be imposed that g1(0) = ˜g1(0) = 1 and g0(0) = 0 (the values of g1and ˜g1at 0 are just for the sake of simplicity when dealing with the limit problem –see Theorems 7and 8–; indeed any other values can be, after rearrangement, easily translated to this situation). Regarding the nonlinearity, we assume that the function f∈C1(R) (cf. Remark 4for a more general setting) and there exist positive constants α1,α2and κ, η ≥0 and p≥2 such that −κ−α1|s|p≤f(s)s≤κ−α2|s|p∀s∈R.(4) f0(s)≤η∀s∈R,(5) From (4) we deduce that there exists β > 0 such that |f(s)| ≤ β(|s|p−1+ 1) ∀s∈R.(6) The structure of this paper is as follows. In the rest of this Section 1, the setting of the problem is established as well as the existence of strong solutions and the regularising effect of the equation. Section 2is devoted to providing abstract results on multi-valued non-autonomous dynamical systems which are essential to prove the existence of pullback attractors in L2(Ω) and H1 0(Ω) in the last two sections. Namely, in Section 3we generalise some results of [6], which guarantee the existence of pullback attractors in L2(Ω) and will be crucial in what follows. In Section 4, 4 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO we show the existence of pullback attractors in H1-norm and analyse their upper semicontinuity with respect to the parameter. Indeed, we prove that both families of attractors, those given in L2(Ω) and H1 0(Ω), converge to the compact global attractor associated to the autonomous problem (P0) in H1-norm when εgoes to 0. Regarding the notation, the inner product in L2(Ω) is represented by (·,·) and its associated norm by |·| (since no confusion arises, this also denotes the Lebesgue measure of a subset of RN). The inner product in H1 0(Ω),given by the product of the gradients in (L2(Ω))N, is represented by ((·,·)),and by k · k the associated norm. The duality product between H−1(Ω) and H1 0(Ω) is denoted by h·,·i and by k · k∗, the norm in H−1(Ω). Identifying L2(Ω) with its dual, the usual chain of dense and compact embeddings H1 0(Ω) ⊂L2(Ω) ⊂H−1(Ω) holds. Observe that, by the Riesz theorem, we can obtain ˜ l∈L2(Ω) with hl, ui(L2(Ω))0,L2(Ω) = (˜ l, u); here on, thanks to the identification (L2(Ω))0≡L2(Ω), we just use linstead of ˜ l, but at the same time we keep the usual notation in the existing previous literature l(u). The duality product between Lp(Ω) and Lq(Ω), where pand qare conjugate exponents, is denoted by (·,·) and the norm in Lp(Ω) is represented by |·|p. Finally, k·kLs(τ,T ;X)denotes the norm in Ls(τ, T ;X) where s≥1 and Xis a separable Banach space. To start with the weak-solution framework, we assume that uτ∈L2(Ω) and h∈L2 loc(R;H−1(Ω)). Definition 1. A weak solution to problem (Pε) is a function uthat belongs to L∞(τ, T ;L2(Ω)) ∩L2(τ, T;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) for all T > τ, with u(τ) = uτ, and such that for all v∈H1 0(Ω) ∩Lp(Ω) d dt(u(t), v) + g1(ε)a(l(u(t)))((u(t), v)) = ˜g1(ε)(f(u(t)), v) + g0(ε)hh(t), vi,(7) where the previous equation must be understood in the sense of D0(τ, ∞). When uis a weak solution to (Pε), making use of the continuity of the function a, (2), (6) and (7), it holds that u0∈L2(τ, T ;H−1(Ω)) + Lq(τ, T;Lq(Ω)) for any T > τ. Therefore, u∈C([τ, ∞); L2(Ω)) and the initial datum in (Pε) makes sense. Furthermore, the following energy equality holds |u(t)|2+ 2g1(ε)Zt s a(l(u(r)))ku(r)k2dr =|u(s)|2+ 2˜g1(ε)Zt s (f(u(r)), u(r))dr + 2g0(ε)Zt s hh(r), u(r)idr (8) for all τ≤s≤t(cf. [22, Th´eor`eme 2, p. 575] or [41, Lemma 3.2, p. 71]). The existence of weak solutions to (Pε) has been proved in [6, Theorem 1] (as commented in the introduction, the lack of Lipschitz character on the function a does not allow to ensure uniqueness). Theorem 1. Assume that (1)–(4) hold and h∈L2 loc(R;H−1(Ω)).Then, for any uτ∈L2(Ω), there exists at least one weak solution to (Pε). Now, in a more regular framework, we will show the regularising effect of the equation and the existence of strong solutions. In order to do that we assume that the function falso fulfils (5) (anyway this assumption can be weakened, see Remark 4for more details), and h∈L2 loc(R;L2(Ω)). ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 5 Definition 2. A strong solution to (Pε) is a weak solution which also belongs to L2(τ, T ;D(−∆)) ∩L∞(τ, T;H1 0(Ω)) for all T > τ. Now we show the regularising effect of the equation for any εand the existence of strong solutions. Theorem 2. Assume that (1)–(5) hold and h∈L2 loc(R;L2(Ω)).Then, for any uτ∈ L2(Ω), each weak solution ubelongs to L∞(τ+, T ;H1 0(Ω)) ∩L2(τ+, T;D(−∆)) for every  > 0and T > τ +. In addition, if the initial datum uτ∈H1 0(Ω),then the weak solutions to (Pε) are in fact strong solutions. Proof. We split the proof into two steps. Step 1. Regularising effect. Fix a weak solution u(·;τ, uτ) to (Pε), for short denoted by u(·).Then, we consider the problem (Pε,u)     dy dt −g1(ε)a(l(u))∆y= ˜g1(ε)f(y) + g0(ε)h(t) in Ω ×(τ, ∞), y= 0 on ∂Ω×(τ, ∞), y(x, τ) = uτ(x) in Ω. Observe that there exists a unique solution to (Pε,u) thanks to the monotonicity of the Laplacian and the assumption (5) made on f(cf. [33, Chapter II]). Thus, more regular (a posteriori) estimates as well as using the Galerkin approximations make complete sense. Moreover, it holds that y=usince usolves (Pε), and (Pε,u) possesses a unique solution. Now, making use of spectral theory and regularity results (cf. [39,32]), consider a sequence {wi}i≥1of eigenfunctions of −∆ in H1 0(Ω),which is a Hilbert basis of L2(Ω).For each integer n≥1,we define the function un(t;τ, uτ) = Pn j=1 ϕnj(t)wj (un(t) for short), which is the local solution to      d dt(un(t), wj)+ g1(ε)a(l(u(t)))((un(t), wj))=(˜g1(ε)f(un(t))+g0(ε)h(t), wj) t∈(τ, ∞), (un(τ), wj)=(uτ, wj), j = 1, . . . , n. (9) Multiplying (9) by ϕnj(t) and summing from j= 1 until n, we deduce 1 2 d dt|un(t)|2+g1(ε)a(l(u(t)))kun(t)k2= ˜g1(ε)(f(un(t)), un(t)) + g0(ε)(h(t), un(t)) a.e. t∈(τ, T ). Integrating the previous expression between τand T, and making use of (1) and (4), we obtain |un(T)|2+2g1(ε)mZT τ kun(t)k2dt ≤ |uτ|2+2˜g1(ε)(T−τ)κ|Ω|+2g0(ε)ZT τ (h(t), un(t))dt. Using the Cauchy inequality, ZT τ kun(t)k2dt ≤1 g1(ε)m|uτ|2+2˜g1(ε)(T−τ)κ|Ω| g1(ε)m+(g0(ε))2 λ1(g1(ε))2m2ZT τ |h(t)|2dt. (10) On the other hand, multiplying (9) by λjϕnj(t),summing from j= 1 until nand using (1), we have 1 2 d dtkun(t)k2+g1(ε)m|−∆un(t)|2≤˜g1(ε)(f(un(t)),−∆un(t))+g0(ε)(h(t),−∆un(t)) 6 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO a.e. t∈(τ, T ). Applying (5) and the Cauchy inequality, it holds d dtkun(t)k2+g1(ε)m|−∆un(t)|2≤2˜g1(ε)ηkun(t)k2+|˜g1(ε)f(0) + g0(ε)h(t)|2 g1(ε)m(11) a.e. t∈(τ, T ). Integrating between sand t, with τ < s ≤t≤T, we obtain kun(t)k2+g1(ε)mZt s | − ∆un(r)|2dr ≤kun(s)k2+ 2˜g1(ε)ηZT τ kun(r)k2dr +1 g1(ε)mZT τ |˜g1(ε)f(0) + g0(ε)h(r)|2dr. Now, integrating w.r.t. sbetween τand t, we have in particular (t−τ)kun(t)k2 ≤(1 + 2˜g1(ε)(T−τ)η)ZT τ kun(r)k2dr +T−τ g1(ε)mZT τ |˜g1(ε)f(0) + g0(ε)h(r)|2dr for all t∈[τ+, T] with ∈(0, T −τ). Then, from this and making use of (10), it holds that the sequence {un}is bounded in L∞(τ+, T;H1 0(Ω)) ∩L2(τ+, T;D(−∆)).Therefore, by the uniqueness of weak solution to (Pε,u), it fulfils (un ∗ * u weakly-star in L∞(τ+ε, T;H1 0(Ω)), un* u weakly in L2(τ+ε, T;D(−∆)). Step 2. Strong solution. Assume that uτ∈H1 0(Ω). We will see that u∈ L∞(τ, T ;H1 0(Ω)) ∩L2(τ, T ;D(−∆)) and actually also u0∈Lq(τ, T;Lq(Ω)) for all T > τ. Integrating (11) between τand t∈[τ, T], we obtain kun(t)k2+g1(ε)mZt τ | − ∆un(r)|2dr ≤kuτk2+ 2˜g1(ε)ηZT τ kun(r)k2dr +1 g1(ε)mZT τ |˜g1(ε)f(0) + g0(ε)h(r)|2dr. Then, taking into account that {un}is bounded in L2(τ, T;H1 0(Ω)), we deduce that {un}is bounded in L∞(τ, T;H1 0(Ω))∩L2(τ, T;D(−∆)).Thanks to the uniqueness of weak solution to (Pε,u), we have (un ∗ * u weakly-star in L∞(τ, T ;H1 0(Ω)), un* u weakly in L2(τ, T ;D(−∆)). On the other hand, since ∂u ∂t −g1(ε)a(l(u))∆u= ˜g1(ε)f(u) + g0(ε)h(t) in Lq(τ, T;Lq(Ω)), it holds that u0∈Lq(τ, T;Lq(Ω)). ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 7 2. Set-valued non-autonomous dynamical systems and pullback attractors. In this section, we provide abstract results on multi-valued non-autonomous dynamical systems (cf. [37,8,36,2]) which are crucial to prove the existence of minimal pullback attractors. Furthermore, results which establish relationships between the families of pullback attractors are also stated (cf. [36]). To set our abstract framework, we consider a metric space (X, dX) and the set R2 d={(t, s)∈R2:t≥s}. In addition, let us denote by P(X) the family of all nonempty subsets of Xand consider a universe D,which is a nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X). Definition 3. A multi-valued map U:R2 d×X7→ P(X) is a multi-valued process on X(also called multi-valued non-autonomous dynamical system) if it fulfils (i) U(τ, τ)x={x} ∀τ∈R∀x∈X, (ii) U(t, τ)x⊂U(t, s)(U(s, τ)x)∀τ≤s≤t∀x∈X, where U(t, τ)W:= [ y∈W U(t, τ)y∀W⊂X. When the relationship established in (ii) is an equality instead of an inclusion, the multi-valued process Uis called strict. Definition 4. A multi-valued process Uon Xis upper-semicontinuous if for all (t, τ)∈R2 d,the mapping U(t, τ) is upper-semicontinuous from Xinto P(X), that is, for each x∈Xand any neighbourhood N(U(t, τ)x) of U(t, τ)x, there exists a neighbourhood Mof xsuch that U(t, τ)y⊂ N(U(t, τ)x) for any y∈ M. Definition 5. A universe Dis called inclusion-closed if given two families b Dand b D0such that b D∈ D and b D0={D0(t) : t∈R} ⊂ P(X) with D0(t)⊂D(t) for all t∈R, it fulfils that b D0∈ D. Now, we consider a family of nonempty sets b D0={D0(t) : t∈R} ⊂ P(X). We do not require any additional condition on these sets such as compactness or boundedness. Definition 6. The family b D0={D0(t) : t∈R}is pullback D-absorbing for a multi-valued process Uif for any t∈Rand b D∈ D, there exists τ(b D, t)≤tsuch that U(t, τ)D(τ)⊂D0(t)∀τ≤τ(b D, t). Definition 7. Given a family b D0={D0(t) : t∈R}, a multi-valued process Uon Xis said to be pullback b D0-asymptotically compact if for any t∈R, every sequence {τn} ⊂ (−∞, t] such that τn→ −∞ and any sequence {xn} ⊂ Xwith xn∈D0(τn) for all n∈N, it fulfils that any sequence {yn}, with each yn∈U(t, τn)xn, is relatively compact in X. Given a universe D,a multi-valued process Uon Xis said to be pullback D- asymptotically compact if it is pullback b D-asymptotically compact for any b D∈ D. Definition 8. A family AD={AD(t) : t∈R} ⊂ P(X) is called the minimal pullback D-attractor for a multi-valued process Uif the following properties are fulfilled: 1. AD(t) is a nonempty compact subset of Xfor any t∈R, 2. ADis pullback D-attracting, i.e. limτ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D and t∈R, where distX(·,·) denotes the Hausdorff semi-distance in Xbetween two subsets of X, 8 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO 3. ADis negatively invariant, i.e. AD(t)⊂U(t, τ)AD(τ) for all (t, τ)∈R2 d, 4. For any family of closed sets b C={C(t) : t∈R}which is pullback D- attracting, the relationship AD(t)⊂C(t) holds for all t∈R. To continue our analysis, we define the omega limit of the family b Din time tby Λ( b D, t) = \ s≤t[ τ≤s U(t, τ)D(τ) X ∀t∈R, where {. . . }Xdenotes the closure in X. Now, we have the main result of this section, which ensures the existence of the minimal pullback D-attractor for a multi-valued process U(cf. [6, Theorem 2]). Theorem 3. Consider an upper-semicontinuous multi-valued process Uwhich has closed values, a pullback D-absorbing family called b D0={D0(t) : t∈R} ⊂ P(X) and also assume that Uis pullback b D0-asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = [ b D∈D Λ( b D, t) X ∀t∈R is the minimal pullback D-attractor and AD(t)⊂D0(t)Xfor all t∈R. In addition, if b D0∈ D, each D0(t)is closed and the universe Dis inclusion-closed, then the family AD∈ D. Moreover, when Uis strict, the family ADis invariant under the multi-valued process U, i.e. AD(t) = U(t, τ)AD(τ)for all (t, τ)∈R2 d. Now, we are going to establish relationships between pullback attractors (cf. [36]), but first we need to introduce some notation. We denote by DX Fthe universe of fixed nonempty bounded subsets of X, i.e. the class of all families b Dof the form b D={D(t) = B:t∈R}, where Bis a fixed nonempty bounded subset of X. Corollary 1. Under the assumptions of Theorem 3, if DX F⊂ D, then the following relationship holds ADX F(t)⊂ AD(t)∀t∈R, where ADX F(t) = [ Bbounded Λ(B, t) X ∀t∈R is the minimal pullback DX F-attractor for the multi-valued process U. In addition, if there exists T∈Rsuch that the set St≤TD0(t)is bounded in X, then ADX F(t) = AD(t)for all t≤T. Thanks to the following result, we can compare two attractors for a process (see [24, Theorem 3.15] for a proof in the single-valued framework). Theorem 4. Suppose that {(Xi, dXi)}i=1,2are two metric spaces such that X1⊂ X2with continuous injection, Diis a universe in P(Xi)for i= 1,2,and D1⊂ D2. Assume that Uis a multi-valued map that acts as a multi-valued process in both cases, i.e. U:R2 d×Xi→ P(Xi)for i= 1,2is a multi-valued process. For each t∈R, Ai(t) = [ b Di∈Di Λi(b Di, t) Xi i= 1,2, ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 9 where the subscript iin the symbol of the omega-limit set Λiis used to denote the dependence on the respective topology. Then, A1(t)⊂ A2(t)for all t∈R. If moreover (i) A1(t)is a compact subset of X1for all t∈R, (ii) for any b D2∈ D2and t∈R, there exist a family b D1∈ D1and a t∗ b D1such that Uis pullback b D1-asymptotically compact, and for any s≤t∗ b D1there exists a τs< s such that U(s, τ)D2(τ)⊂D1(s)for all τ≤τs, then A1(t) = A2(t)for all t∈R. 3. Previous results on the asymptotic behaviour in L2-norm. In [6] the existence of pullback attractors to (Pε) in L2(Ω) is analysed. Namely, in what follows we will recall the main results that guarantee the existence of these families. This is the first step in order to state our regularity results in H1 0(Ω) in Section 4. Observe that the results are provided without proofs since (Pε) is a slight generalization of the one analysed in [6]. However, we include the (adapted) statements here for the sake of clarity when reading the next section. Thanks to the existence of weak solutions to (Pε) (cf. [6, Theorem 1]), we can define a multi-valued map Uε:R2 d×L2(Ω) → P(L2(Ω)) as Uε(t, τ)uτ={u(t) : u∈Φε(τ, uτ)}, τ ≤t, uτ∈L2(Ω), where Φε(τ, uτ) denotes the set of weak solutions to (Pε) in [τ, ∞) with initial datum uτ∈L2(Ω). The following result is the natural generalization of [6, Lemma 1, Proposition 2] to this setting. Proposition 1. Assume that (1)–(4) hold and h∈L2 loc(R;H−1(Ω)).Then, Uε: R2 d×L2(Ω) → P(L2(Ω)) is a strict upper-semicontinuous multi-valued process with closed values for all ε∈[0,1]. From now on, for any µ > 0, the class of all families of nonempty subsets b D= {D(t) : t∈R}⊂P(L2(Ω)) such that lim τ→−∞ eµτ sup v∈D(τ) |v|2!= 0 is denoted by DL2 µ. It is worth noting that DL2 F⊂ DL2 µand DL2 µis inclusion-closed. From now on, we assume a condition to simplify the exposition and the form of the limit problem (P0), namely g1(0) = 1.(12) To prove the existence of a pullback absorbing family, we assume that there exists ¯µ∈(0,2λ1m) such that Z0 −∞ e¯µskh(s)k2 ∗ds < ∞.(13) Remark 1. From the continuity of g1and (12) it is immediate to deduce that there exists ¯ε∈(0,1] such that ¯µ < 2g1(ε)λ1mfor all ε∈[0,¯ε].Indeed, this last condition will be used in the sequel to construct the absorbing families in suitable universes DL2 µε. Furthermore, it is consistent with the final goal of studying the limit of problems (Pε) when εgoes to 0. In what follows the parameter µεis taken in [¯µ, 2g1(ε)λ1m) since (13) also holds with the weight e¯µs replaced by eµεsand DL2 ¯µ⊂ DL2 µε,so a larger class of objects will be attracted. 16 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO a.e. ξ > τ. Integrating between rand swith τ≤r−1≤s≤r, we obtain in particular kun(r)k2≤kun(s)k2+ 2˜g1(ε)ηZr r−1 kun(ξ)k2dξ +1 g1(ε)mZr r−1 |˜g1(ε)f(0) + g0(ε)h(ξ)|2dξ. Integrating the last inequality w.r.t. sbetween r−1 and r, kun(r)k2≤(1 + 2˜g1(ε)η)Zr r−1 kun(s)k2ds +1 g1(ε)mZr r−1 |˜g1(ε)f(0) + g0(ε)h(ξ)|2dξ for all τ≤r−1. Therefore, making use of the estimate on the solutions given by (ρε 2)ext, it fulfils for any n≥1 kun(r;τ, uτ)k2≤˜ρε 1(t)∀r∈[t−2, t]∀uτ∈D(τ)∀τ≤τ3(b D, t),(30) where ˜ρε 1(t) is given in the statement. Now, taking limit inferior in (30) and using the well-known fact that unconverge to u(·;τ, uτ)∈C([t−2, t]; H1 0(Ω)) weakly-star in L∞(t−2, t;H1 0(Ω)) (cf. Theorem 2), the first inequality in (26) holds. Then integrating between r−1 and rin (29), we obtain in particular for any n≥1 Zr r−1 | − ∆un(ξ)|2dξ ≤1 g1(ε)mkun(r−1)k2+ 2˜g1(ε)ηZr r−1 kun(ξ)k2dξ +1 g1(ε)mZr r−1 |˜g1(ε)f(0) + g0(ε)h(ξ)|2dξ for all τ≤r−1.Then, Zr r−1 | − ∆un(ξ)|2dξ ≤˜ρε 2(t)∀r∈[t−1, t]∀uτ∈D(τ)∀τ≤τ3(b D, t),(31) where ˜ρε 2(t) is given in the statement. Then, taking limit inferior in (31) and bearing in mind that unconverge to uweakly in L2(r−1, r;H2(Ω) ∩H1 0(Ω)) for all r∈[t−1, t] (cf. Theorem 2), the second inequality in (26) holds. Now, taking into account that fsatisfies (16) (see also (17)) together with the previous estimates, it holds Zr r−1 |f(un(ξ))|2dξ ≤2C2|Ω|+ 2C2(CH1 0(N))2˜ b(CH2(N))2ˆ b(˜ρε 1(t))ˆ b(˜ρε 2(t))˜ b(32) for all r∈[t−1, t], uτ∈D(τ) and τ≤τ3(b D, t), where ˆ b= (1−θ)(γ+1), ˜ b=θ(γ+1), CH1 0(N) and CH2(N) are the constants of the continuous embeddings of H1 0(Ω) and H2(Ω) into Lp-spaces respectively and θ∈[0,1]. Finally, observe that for all τ≤r−1 Zr r−1 |u0 n(ξ)|2dξ ≤3(g1(ε))2Zr r−1 | − a(l(un(ξ)))∆un(ξ)|2dξ + 3(˜g1(ε))2Zr r−1 |f(un(ξ))|2dξ + 3(g0(ε))2Zr r−1 |h(ξ)|2dξ. Then, using (28), (31) and (32), we obtain for any n≥1 Zr r−1 |u0 n(ξ)|2dξ ≤˜ρε 3(t)∀r∈[t−1, t]∀uτ∈D(τ)∀τ≤τ3(b D, t), ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 17 where ˜ρε 3is given in the statement. Finally, taking limit inferior in the above expression and using that u0 nconverge to u0(·;τ, uτ) weakly in L2(r−1, r;L2(Ω)) for all r∈[t−1, t], we deduce the last inequality in (26). Now, to prove the pullback asymptotic compactness of Uεin H1 0(Ω) for the universe DL2,H1 0 µε, we apply an energy method which relies on the continuity of solutions (see [27,35,36,24] for more details). Proposition 6. Under the assumptions of Proposition 5, there exists ¯ε∈(0,1] such that for any ε∈[0,¯ε]and µε∈[¯µ, 2g1(ε)λ1m),the process Uε:R2 d×H1 0(Ω) → P(H1 0(Ω)) is pullback DL2,H1 0 µε-asymptotically compact. Proof. According to Proposition 5let us fix ε∈[0,¯ε], µε∈[¯µ, 2g1(ε)λ1m), t∈R, a family b DH1 0∈ DL2,H1 0 µε,sequences {τn} ⊂ (−∞, t −3] with τn→ −∞ and {un τ} with un τ∈D(τn) for all n. We aim to prove that any sequence {yn},where yn∈ Uε(t, τn)un τfor all n, is relatively compact in H1 0(Ω).In fact, since yn∈Uε(t, τn)un τ, there exists un∈Φε(τn, un τ) such that yn=un(t).Therefore, we will show that the sequence {un(t)}is relatively compact in H1 0(Ω). As a consequence of Lemma 2, there exists τ3(b D, t)< t −3,such that τn≤ τ3(b D, t) for all n≥n1,the sequence {un}n≥n1is bounded in L∞(t−2, t;H1 0(Ω)) ∩ L2(t−2, t;H2(Ω)),and {−a(l(un))∆un}n≥n1,{f(un)}n≥n1and {(un)0}n≥n1are bounded in L2(t−2, t;L2(Ω)).Then, using the Aubin-Lions lemma, there exists u∈L∞(t−2, t;H1 0(Ω)) ∩L2(t−2, t;H2(Ω) ∩H1 0(Ω)) with u0∈L2(t−2, t;L2(Ω)), such that for a subsequence (relabeled the same) it holds                            un∗ * u weakly-star in L∞(t−2, t;H1 0(Ω)), un* u weakly in L2(t−2, t;H2(Ω) ∩H1 0(Ω)), (un)0* u0weakly in L2(t−2, t;L2(Ω)), un→ustrongly in L2(t−2, t;H1 0(Ω)), un(s)→u(s) strongly in H1 0(Ω) a.e. s∈(t−2, t), f(un)* f(u) weakly in L2(t−2, t;L2(Ω)), −a(l(un))∆un*−a(l(u))∆uweakly in L2(t−2, t;L2(Ω)), (33) where the last two convergences have been identified using [33, Lemma 1.3, p. 12]. Then we deduce that u∈C([t−2, t]; H1 0(Ω)) and fulfils (7) in the interval (t−2, t). Moreover, since {(un)0}n≥n1is bounded in L2(t−2, t;L2(Ω)), it satisfies that {un}n≥n1is equicontinuous in L2(Ω) on [t−2, t].From this and taking into account that {un}n≥n1is bounded in L∞(t−2, t;H1 0(Ω)) and the compactness of the embedding H1 0(Ω) ,→L2(Ω), applying the Ascoli-Arzel´a Theorem we obtain un→ustrongly in C([t−2, t]; L2(Ω)).(34) On the other hand, using that {un}n≥n2is bounded in C([t−2, t]; H1 0(Ω)),we have that for any sequence {sn} ⊂ [t−2, t] with sn→s∗, un(sn)* u(s∗) weakly in H1 0(Ω),(35) where (34) has been used to identify the weak limit. If we prove un→ustrongly in C([t−1, t]; H1 0(Ω)),(36) 18 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO in particular, we will deduce that the sequence {un(t)}is relatively compact in H1 0(Ω). To that end, we argue by contradiction. We suppose that there exist ε > 0, a sequence {tn} ⊂ [t−1, t],without loss of generality converging to some t∗,with kun(tn)−u(t∗)k ≥ ε∀n≥1.(37) From (35), it holds ku(t∗)k ≤ lim inf n→∞ kun(tn)k. It is not difficult to prove, making use of the Galerkin approximations, that kun(s)k2≤kun(r)k2−2˜g1(ε)ηZs r kun(ξ)k2dξ −1 2g1(ε)mZs r |˜g1(ε)f(0) + g0(ε)h(ξ)|2dξ (38) for all t−2≤r≤s≤t. Then, we define the following continuous functions on [t−2, t] Jn(s) = kun(s)k2−2˜g1(ε)ηZs t−2 kun(r)k2dr−1 2g1(ε)mZs t−2 |˜g1(ε)f(0) + g0(ε)h(r)|2dr, J(s) = ku(s)k2−2˜g1(ε)ηZs t−2 ku(r)k2dr−1 2g1(ε)mZs t−2 |˜g1(ε)f(0) + g0(ε)h(r)|2dr. Observe that thanks to (38), all the functions Jnare non-increasing on the interval [t−2, t].In addition, taking into account the definition of Jnand (33), it holds Jn(s)→J(s) a.e. s∈(t−2, t). Hence, there exists a sequence {˜ tk} ⊂ (t−2, t∗) such that ˜ tk→t∗when k→ ∞ and lim n→∞ Jn(˜ tk) = J(˜ tk)∀k≥1. Consider fixed  > 0. Since the function Jis continuous on [t−2, t], there exists k()≥1 such that |J(˜ tk)−J(t∗)|< 2∀k≥k(). Now, we consider n()≥1 such that tn≥˜ tk()and |Jn(˜ tk())−J(˜ tk())|< 2∀n≥n(). Since all the functions Jnare non-increasing, for all n≥n() Jn(tn)−J(t∗)≤Jn(˜ tk())−J(t∗) ≤ |Jn(˜ tk())−J(t∗)| ≤ |Jn(˜ tk())−J(˜ tk())|+|J(˜ tk())−J(t∗)| < 2+ 2=. Then, lim supn→∞ Jn(tn)≤J(t∗). Thus, it satisfies that lim supn→∞ kun(tn)k ≤ ku(t∗)kwhich, together with (35), allow us to prove that {un(tn)}converges to u(t∗) strongly in H1 0(Ω), in contradiction with (37). Therefore, (36) holds. The following result shows the existence of pullback attractors in H1 0(Ω) as well as some relationships between them. ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 19 Theorem 6. Assume that (1)–(5), (12) and (16) hold, and that h∈L2 loc(R;L2(Ω)) satisfies (13) for some ¯µ∈(0,2λ1m). Then, there exists ¯ε∈(0,1] such that for any ε∈[0,¯ε]and µε∈[¯µ, 2g1(ε)λ1m),there exist the minimal pullback DH1 0 F-attractor Aε DH1 0 F and the minimal pullback DL2,H1 0 µε-attractor Aε DL2,H1 0 µε for the multi-valued process Uε:R2 d×H1 0(Ω) → P(H1 0(Ω)). Furthermore, it fulfils Aε DH1 0 F (t)⊂ Aε DL2 F (t)⊂ Aε DL2 µε (t) = Aε DL2,H1 0 µε (t)∀t∈R∀ε∈[0,¯ε].(39) In particular, for any b D∈ DL2 µε, the following pullback attraction in H1 0(Ω) holds lim τ→−∞ distH1 0(Uε(t, τ)D(τ),Aε DL2 µε (t)) = 0 ∀t∈R.(40) Finally, if there exists some ˜µ∈(0,2λ1m)such that sup s≤0e−˜µs Zs −∞ e˜µr|h(r)|2dr<∞,(41) then there exists ˜ε∈(0,1] such that Aε DH1 0 F (t) = Aε DL2 F (t) = Aε DL2 µ (t) = Aε DL2,H1 0 µ (t)∀t∈R∀ε∈[0,˜ε]. In addition, lim τ→−∞ distH1 0(Uε(t, τ)B, Aε DL2 F (t)) = 0 ∀t∈R∀B∈ DL2 F∀ε∈[0,˜ε].(42) Proof. As in the previous results, for ε∈[0,¯ε] and µε∈[¯µ, 2g1(ε)λ1m),the existence of Aε DH1 0 F and Aε DL2,H1 0 µε is a consequence of Corollary 1. Indeed, the process Uεis upper-semicontinuous with closed values (cf. Proposition 4), the relation DH1 0 F⊂ DL2,H1 0 µεis fulfilled, and the existence of an absorbing family (cf. Proposition 5) and the asymptotic compactness (cf. Proposition 6) hold. The chain of inclusions (39) follows from Corollary 1and Theorem 4. In fact, the equality for all t∈Rbetween Aε DL2 µε (t) and Aε DL2,H1 0 µε (t) is also due to Theorem 4, using Proposition 5. Then, (40) obviously holds. After (41), the equality Aε DH1 0 F (t) = Aε DL2 F (t) is again due to Theorem 4, making use of the first estimate appearing in Lemma 2. Therefore, (42) is straightforward. Till now, in the previous results the problem (P0) could be non-autonomous. However we aim to consider (as done in [6]) the case of (P0) being autonomous, approached by perturbed problems coming from some noise that affects the coefficients and in particular it includes time-dependent forces (i.e. h). So in the sequel we assume that g0(0) = 0.(43) Furthermore, for the sake of simplicity in the formulation of the limit problem, we also assume that ˜g1(0) = 1.(44) Remark 3. (i) Under the new assumption (43), problem (P0) becomes autonomous. Since the above results also hold for (P0), its associated pullback attractors are in fact the global compact attractor A0 DL2 F in L2(Ω) for the multi-valued semiflow 20 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO S, where S(t−τ) = U0(t, τ). Namely, it can be seen as a pullback attractor for the universes DL2 Fand DL2 µ0with µ0= 2λ1m(cf. Proposition 2). Indeed, A0 DL2 µ0 (t) = A0 DL2 F holds for all t∈R. (ii) Analogously there exists the compact global attractor A0 DH1 0 F in H1 0(Ω). This set can be seen as pullback attractor for the universes DH1 0 Fand DL2,H1 0 µ0. Namely, A0 DL2,H1 0 µ0 (t) = A0 DH1 0 F for all t∈R. Finally, the upper semicontinuous behaviour of the pullback attractors Aε DL2 µε (t) = Aε DL2,H1 0 µε (t) in H1-norm as εgoes to 0 for all t∈Ris analysed. As done in [6], to prove these properties we first establish the following continuity (in ε) result of solutions to (Pε) toward solutions of the limit problem (P0)     ∂u ∂t −a(l(u))∆u=f(u) in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x) in Ω. Theorem 7. Assume that (1)–(5), (12), (16), (43) and (44) hold, h∈L2 loc(R;L2(Ω)) and consider sequences {εn}with limnεn= 0 and {un τ} ⊂ L2(Ω) such that un τ* uτ weakly in L2(Ω). Then, there exist a subsequence of {un τ}(relabeled the same), a sequence {uεn},with uεn∈Φεn(τ, un τ),and u0∈Φ0(τ, uτ)such that uεn(t)→u0(t)strongly in H1 0(Ω) for all t > τ. (45) Proof. Let {un τ} ⊂ L2(Ω) be a sequence such that un τ* uτweakly in L2(Ω). Consider arbitrary values T > τ and δ∈(0, T −τ). Consider uεna weak solution to (Pεn) in [τ, T]. Making use of the energy equality and (1), we obtain 1 2 d ds|uεn(s)|2+g1(εn)mkuεn(s)k2≤˜g1(εn)(f(uεn(s)), uεn(s))+g0(εn)hh(s), uεn(s)i a.e. s∈[τ, T ]. Now, define γ:= minn{g1(εn)} ∈ (0,1].Then, taking this into account together with (4) and the Cauchy inequality, we have d ds|uεn(s)|2+γmkuεn(s)k2+ 2α2˜g1(εn)|uεn(s)|p p≤2˜g1(εn)κ|Ω|+(g0(εn))2kh(s)k2 ∗ γm a.e. s∈[τ, T ]. Then, integrating between τand twith t∈[τ, T],we deduce |uεn(t)|2+γm Zt τ kuεn(s)k2ds + 2α2˜g1(εn)Zt τ |uεn(s)|p pds ≤|un τ|2+ 2˜g1(εn)κ|Ω|(T−τ) + (g0(εn))2 γm ZT τ kh(s)k2 ∗ds. Therefore, {uεn}is bounded in L∞(τ, T;L2(Ω))∩L2(τ, T;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)). In addition, since each uεn∈C([τ, T]; L2(Ω)) for all n, it holds |uεn(t)| ≤ C∞∀t∈[τ, T]∀n≥1, where C∞is a positive constant independent of εn. Now, as l∈L2(Ω) and a∈ C(R; [0,∞)), there exists MC∞>0 such that a(l(uεn(t))) ≤MC∞∀t∈[τ, T ]∀n≥1. ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 21 Then, it is standard to deduce that there exist a subsequence of {uεn}(relabeled the same) and u0∈C([τ, T]; L2(Ω)) ∩L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)), solution to (P0) with u0(τ) = uτ. On the other hand, making use of the strong energy equality (18) for the Galerkin approximations to problems (Pεn,uεn), applying (1) and (5), and passing to the limit, we obtain kuεn(t)k2+g1(εn)mZt τ+δ | − ∆uεn(r)|2dr ≤ kuεn(τ+δ)k2 + 2˜g1(εn)ηZt τ+δ kuεn(r)k2dr +1 g1(εn)mZt τ+δ |˜g1(εn)f(0) + g0(εn)h(r)|2dr for all δ∈(0, t −τ). Taking this into account together with Theorem 2, the sequence {uεn}is bounded in L∞(τ+δ, T;H1 0(Ω))∩L2(τ+δ, T;H2(Ω) ∩H1 0(Ω)). Then from (20), the sequence {−a(l(uεn))∆uεn}is bounded in L2(τ+δ, T;L2(Ω)). From (16) and the previous boundedness of {uεn}, we deduce that {f(uεn)}is bounded in L2(τ+δ, T;L2(Ω)). Finally, bearing in mind the above estimates and the problems (Pεn) we obtain that {(uεn)0}is bounded in L2(τ+δ, T;L2(Ω)). Then, we gain that the limit u0∈L∞(τ+δ, T;H1 0(Ω))∩L2(τ+δ, T;H2(Ω)∩H1 0(Ω)) with (u0)0∈L2(τ+δ, T;L2(Ω)). Now using the Aubin-Lions lemma, there exists a subsequence of {uεn}(relabeled the same) such that                              uεn∗ * u0weakly-star in L∞(τ+δ, T;H1 0(Ω)), uεn* u0weakly in L2(τ+δ, T;H2(Ω)), uεn→u0strongly in L2(τ+δ, T;H1 0(Ω)), uεn(t)→u0(t) strongly in H1 0(Ω) a.e. t∈(τ+δ, T), (uεn)0*(u0)0weakly in L2(τ+δ, T;L2(Ω)), ˜g1(εn)f(uεn)* f(u0) weakly in L2(τ+δ, T;L2(Ω)), −g1(εn)a(l(uεn))∆uεn*−a(l(u0))∆u0weakly in L2(τ+δ, T;L2(Ω)), (46) where the limits of the last two convergences have been identified using [33, Lemma 1.3, p. 12]. In fact, we may now repeat the arguments in the intervals (τ+δ/2, T +1), (τ+δ/3, T + 2), etcetera, and making use of a diagonal argument, (46) holds in (τ+˜ δ, ˜ T) for all ˜ T > τ and any ˜ δ∈(0,˜ T−τ). Now, we complete the proof by showing (45). Given t > τ, consider T > t and δ∈(0, T −τ). On the one hand, taking into account that the sequences {uεn}and {(uεn)0} are bounded in C([τ+δ, T]; H1 0(Ω)) and L2(τ+δ, T;L2(Ω)) respectively, and the compactness of the embedding H1 0(Ω) ,→L2(Ω), the Ascoli-Arzel`a Theorem implies that uεn→u0strongly in C([τ+δ, T]; L2(Ω)).(47) In fact, since {uεn}is bounded in C([τ+δ, T]; H1 0(Ω)),we obtain uεn(s)* u0(s) weakly in H1 0(Ω) ∀s∈[τ+δ, T],(48) where (47) has been used to identify the weak limit. On the other hand, again from the energy equality (18) for the Galerkin approximations for (Pεn,uεn), applying (1) and (5) and passing to the limit, we have for 22 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO all τ+δ≤r≤s≤T kuεn(s)k2≤kuεn(r)k2+ 2˜g1(εn)ηZs τ+δ kuεn(ξ)k2dξ +1 2g1(εn)mZs τ+δ |˜g1(εn)f(0) + g0(εn)h(ξ)|2dξ. (49) Now, we define the following continuous functions on [τ+δ, T] Jεn(s) =kuεn(s)k2−2˜g1(εn)ηZs τ+δ kuεn(ξ)k2dξ −1 2g1(εn)mZs τ+δ |˜g1(εn)f(0) + g0(εn)h(ξ)|2dξ, J0(s) =ku0(s)k2−2ηZs τ+δ ku0(ξ)k2dξ −(f(0))2|Ω|[s−(τ+δ)] 2m. Observe that from (49) we deduce that all the functions Jnare non-increasing on [τ+δ, T]. Furthermore, since uεn(t)→u0(t) strongly in H1 0(Ω) a.e. t∈(τ+δ, T), Jis continuous in [τ+δ, T] and all Jnare non-increasing in [τ+δ, T], it holds Jεn(s)→J0(s)∀s∈[τ+δ, T]. From this, we deduce lim n→∞ kuεn(s)k2=ku0(s)k2∀s∈[τ+δ, T]. Taking this into account, together with (48), (45) holds. In order to prove the upper semicontinuous behaviour of attractors in H1 0(Ω) we introduce a last condition relating some terms involved in the formula for Rε L2when εgoes to 0, namely we assume that lim sup ε→0 (g0(ε))2 2g1(ε)m−λ−1 1µε <∞,(50) where µεare chosen in [¯µ, 2g1(ε)λ1m). Observe that in [6] we did not specify how to choose µε. Actually we just said that they could be taken equal to µε0. In this paper, condition (50) provides how close to zero the amount 2g1(ε)m−λ−1 1µεcan be such that the whole fraction in (50) is O(1). This fact will be essential in the proof of our main result. Theorem 8. Assume that (1)–(5), (12), (16), (43), (44) and (50) hold, and h∈ L2 loc(R;L2(Ω)) satisfies (13) for some ¯µ∈(0,2λ1m). Then, there exists ¯ε∈(0,1] such that for any ε∈[0,¯ε]and any µε∈[¯µ, 2g1(ε)λ1m),the family {Aε DL2 µε (t)}ε∈(0,¯ε] converges upper semicontinuously to A0 DL2 F in H1 0(Ω) as εgoes to 0, i.e. lim ε→0distH1 0(Aε DL2 µε (t),A0 DL2 F )=0 ∀t∈R.(51) Proof. As in previous results, denote ¯ε∈(0,1] such that ¯µ < 2g1(ε)λ1mfor any ε∈[0,¯ε] and take µε∈[¯µ, 2g1(ε)λ1m).Without loss of generality, from (50) consider a constant C > 0 such that (g0(ε))2 2g1(ε)m−λ−1 1µε < C ∀ε∈[0,¯ε]. ROBUSTNESS OF ATTRACTORS IN H1-NORM FOR NONLOCAL PROBLEMS 23 We prove (51) arguing by contradiction. Suppose that there exist  > 0, t∈R and a sequence {εn}n≥1⊂(0,¯ε] with limn→∞ εn= 0 such that distH1 0(Aεn DL2 µεn (t),A0 DL2 F )>  ∀n∈N. Since the pullback attractors are negatively invariant (cf. Definition 8), there exists a sequence of solutions {uεn}n≥1with uεn(t)∈ Aεn DL2 µεn (t) such that dH1 0(uεn(t),A0 DL2 F )>  ∀n∈N,(52) where the distance in H1 0(Ω) makes complete sense thanks to the regularising effect of the equation (cf. Theorem 2) and the fact that A0 DL2 F =A0 DH1 0 F thanks to the cited regularising effect. Observe that Aεn DL2 µεn (t)⊂Dεn 0,H1 0(t) (cf. Theorem 2and Proposition 5) for all n∈Nand t∈R, and Dεn 0,H1 0(t) = BL2(0,(Rεn L2(t))1/2)∩H1 0(Ω), where we remind that Rεn L2(t) = 1 + 2˜g1(εn)κ|Ω| µεn +(g0(εn))2e−µεnt 2g1(εn)−λ−1 1µεnZt −∞ eµεnskh(s)k2 ∗ds. Taking into account (50), µεn∈[¯µ, 2g1(εn)λ1m),and denoting f M:= max[0,1] |˜g1(·)|, we obtain that Rε L2(t)≤R0 L2(t) := 1+2κf M|Ω| ¯µ+Ce e C(−t)Zt −∞ e e C(s)kh(s)k2 ∗ds ∀t∈R∀ε∈(0,¯ε], where e C(s) = ¯µs if s≤0, 2λ1ms if s > 0. In particular, observe that for t < 0 we have that R0 L2(t) = 1 + 2κf M|Ω| ¯µ+Ce−2λ1mt Zt −∞ e¯µskh(s)k2 ∗ds, whence it is obvious that the family e D0 H1 0={D0 H1 0(t) := BL2(0,(R0 L2(t))1/2)∩H1 0(Ω) : t∈R}∈DL2,H1 0 2λ1m. From the above inclusions and definition we deduce that Aεn DL2 µεn (t)⊂D0 H1 0(t)∀t∈R∀n≥1.(53) On the other hand, since e D0 H1 0belongs to DL2,H1 0 2λ1m, there exists τ(t, e D0 H1 0, )< t such that distH1 0(U0(t, τ)D0 H1 0(τ),A0 DL2 F )≤ 2∀τ≤τ(t, e D0 H1 0, ).(54) From (53), we deduce that the sequence {uεn(τ(t, e D0 H1 0, ))}n≥1is bounded and possesses a subsequence (relabeled the same) such that uεn(τ(t, e D0 H1 0, )) * uτ weakly in L2(Ω). Now, applying Theorem 7, we deduce that there exists u0∈Φ0(τ, uτ) and a subsequence of {εn}n≥1(relabeled the same) such that (45) holds for all t>τ(t, e D0 H1 0, ). Thus, we deduce that there exists n0≥1 such that kuεn(t)−u0(t)k ≤  2∀n≥n0.(55) 24 T. CARABALLO, M. HERRERA-COBOS AND P. MAR´ IN-RUBIO Now, using (54) and (55), we have dH1 0(uεn(t),A0 DL2 F )≤dH1 0(uεn(t), u0(t)) + dH1 0(u0(t),A0 DL2 F ) ≤ 2+ 2=∀n≥n0, which is contradictory with (52). Remark 4. (i) The results concerning weak solutions and L2-attractors only make use of fcontinuous and fulfilling (4) (cf. [6]). However, in the strong framework, assumption (5) is used. Therefore, to avoid confusion in the exposition this regularity has been imposed from the beginning. Nevertheless, this last condition (5) can be replaced by the weaker one (f(s)−f(r))(s−r)≤η(s−r)2∀s, r ∈R, with fjust continuous. To that end, just simply considering mollifiers ρδ, which implies that fδ=ρδ∗ffulfils (5), and compactness arguments. (ii) The values that γtakes in assumption (16) are larger if the interpolation result is applied not only to L∞(τ, T ;H1 0(Ω)) ∩L2(τ, T ;H2(Ω) ∩H1 0(Ω)), but also to L∞(τ, T ;H1 0(Ω))∩Lp(τ, T;Lp(Ω)) and L2(τ, T;H2(Ω)∩H1 0(Ω))∩Lp(τ, T;Lp(Ω)). 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