ROBUSTNESS OF TIME-DEPENDENT ATTRACTORS IN H 1 -NORM FOR NONLOCAL PROBLEMS

. In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H 1 -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 a is a continuous function and l ∈ (L 2 (Ω)) , i.e. l(u) = l g (u) = Ω g(x)u(x)dx.
From a biological point of view, the function u might represent the density of a population. Additional assumptions could be imposed on the function a to 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 a to 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, 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 L 2 (Ω).
If f also 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-dynamicalsystem 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 L 2 (Ω) and H 1 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 L 2 (Ω) is analysed in a multi-valued framework for the non-autonomous nonlocal reaction-diffusion problem where ε ∈ [0, 1] and no locally Lipschitz assumption on the function a or monotonicity on the nonlinearity f are imposed (so lack of uniqueness). Moreover, the upper semicontinuous behaviour of attractors in L 2 -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 ε goes 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 H 1 0 (Ω) as well as their upper semicontinuous behaviour in H 1 -norm. According to [6,Section 6] we consider the parameterized family (ε ∈ [0, 1]) of non-autonomous nonlocal reaction-diffusion problems where Ω is a bounded open subset of R N of class C 1,1 , τ ∈ R, the nonlocal diffusion term fulfils that there exists m > 0 such that l ∈ (L 2 (Ω)) .
(2) Concerning the family of perturbed coefficients, varying with the parameter ε, suppose that Actually, at some stages of the paper -not from the very beginning-it will be imposed that g 1 (0) =g 1 (0) = 1 and g 0 (0) = 0 (the values of g 1 andg 1 at 0 are just for the sake of simplicity when dealing with the limit problem -see Theorems 7 and 8-; indeed any other values can be, after rearrangement, easily translated to this situation).
Regarding the nonlinearity, we assume that the function f ∈ C 1 (R) (cf. Remark 4 for a more general setting) and there exist positive constants α 1 , α 2 and κ, η ≥ 0 and p ≥ 2 such that (4) From (4) we deduce that there exists β > 0 such that 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 2 is devoted to providing abstract results on multi-valued non-autonomous dynamical systems which are essential to prove the existence of pullback attractors in L 2 (Ω) and H 1 0 (Ω) in the last two sections. Namely, in Section 3 we generalise some results of [6], which guarantee the existence of pullback attractors in L 2 (Ω) and will be crucial in what follows. In Section 4, we show the existence of pullback attractors in H 1 -norm and analyse their upper semicontinuity with respect to the parameter. Indeed, we prove that both families of attractors, those given in L 2 (Ω) and H 1 0 (Ω), converge to the compact global attractor associated to the autonomous problem (P 0 ) in H 1 -norm when ε goes to 0.
Regarding the notation, the inner product in L 2 (Ω) is represented by (·, ·) and its associated norm by | · | (since no confusion arises, this also denotes the Lebesgue measure of a subset of R N ). The inner product in H 1 0 (Ω), given by the product of the gradients in (L 2 (Ω)) N , is represented by ((·, ·)), and by · the associated norm. The duality product between H −1 (Ω) and H 1 0 (Ω) is denoted by ·, · and by · * , the norm in H −1 (Ω). Identifying L 2 (Ω) with its dual, the usual chain of dense and compact embeddings H 1 0 (Ω) ⊂ L 2 (Ω) ⊂ H −1 (Ω) holds. Observe that, by the Riesz theorem, we can obtainl ∈ L 2 (Ω) with l, u (L 2 (Ω)) ,L 2 (Ω) = (l, u); here on, thanks to the identification (L 2 (Ω)) ≡ L 2 (Ω), we just use l instead ofl, but at the same time we keep the usual notation in the existing previous literature l(u). The duality product between L p (Ω) and L q (Ω), where p and q are conjugate exponents, is denoted by (·, ·) and the norm in L p (Ω) is represented by | · | p . Finally, · L s (τ,T ;X) denotes the norm in L s (τ, T ; X) where s ≥ 1 and X is a separable Banach space.
When u is a weak solution to (P ε ), making use of the continuity of the function a, (2), (6) and (7), it holds that u ∈ L 2 (τ, T ; H −1 (Ω)) + L q (τ, T ; L q (Ω)) for any T > τ . Therefore, u ∈ C([τ, ∞); L 2 (Ω)) and the initial datum in (P ε ) makes sense. Furthermore, the following energy equality holds The existence of weak solutions to (P ε ) 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 ∈ L 2 loc (R; H −1 (Ω)). Then, for any u τ ∈ L 2 (Ω), there exists at least one weak solution to (P ε ). 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 f also fulfils (5) (anyway this assumption can be weakened, see Remark 4 for more details), and h ∈ L 2 loc (R; L 2 (Ω)).

Definition 2.
A strong solution to (P ε ) is a weak solution which also belongs to L 2 (τ, T ; D(−∆)) ∩L ∞ (τ, T ; H 1 0 (Ω)) for all T > τ . Now we show the regularising effect of the equation for any ε and the existence of strong solutions.
Proof. We split the proof into two steps.
On the other hand, since it holds that u ∈ L q (τ, T ; L q (Ω)).

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, d X ) and the set In addition, let us denote by P(X) the family of all nonempty subsets of X and consider a universe D, which is a nonempty class of families parameterized in time D = {D(t) : t ∈ R} ⊂ P(X).
When the relationship established in (ii) is an equality instead of an inclusion, the multi-valued process U is called strict. Now, we consider a family of nonempty sets D 0 = {D 0 (t) : t ∈ R} ⊂ P(X). We do not require any additional condition on these sets such as compactness or boundedness.
Definition 7. Given a family D 0 = {D 0 (t) : t ∈ R}, a multi-valued process U on X is said to be pullback D 0 -asymptotically compact if for any t ∈ R, every sequence {τ n } ⊂ (−∞, t] such that τ n → −∞ and any sequence {x n } ⊂ X with x n ∈ D 0 (τ n ) for all n ∈ N, it fulfils that any sequence {y n }, with each y n ∈ U (t, τ n )x n , is relatively compact in X. Given a universe D, a multi-valued process U on X is said to be pullback Dasymptotically compact if it is pullback D-asymptotically compact for any D ∈ D.
is called the minimal pullback D-attractor for a multi-valued process U if the following properties are fulfilled: where dist X (·, ·) denotes the Hausdorff semi-distance in X between two subsets of X, To continue our analysis, we define the omega limit of the family D in time t by denotes 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 U which has closed values, a pullback D-absorbing family called D 0 = {D 0 (t) : t ∈ R} ⊂ P(X) and also assume that U is pullback D 0 -asymptotically compact. Then, the family d . Now, we are going to establish relationships between pullback attractors (cf. [36]), but first we need to introduce some notation. We denote by D X F the universe of fixed nonempty bounded subsets of X, i.e. the class of all families D of the form D = {D(t) = B : t ∈ R}, where B is a fixed nonempty bounded subset of X.
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).
Assume that U is a multi-valued map that acts as a multi-valued process in both cases, i.e. U : where the subscript i in the symbol of the omega-limit set Λ i is used to denote the dependence on the respective topology. Then, If moreover (i) A 1 (t) is a compact subset of X 1 for all t ∈ R, (ii) for any D 2 ∈ D 2 and t ∈ R, there exist a family D 1 ∈ D 1 and a t * D1 such that U is pullback D 1 -asymptotically compact, and for any 3. Previous results on the asymptotic behaviour in L 2 -norm. In [6] the existence of pullback attractors to (P ε ) in L 2 (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 H 1 0 (Ω) in Section 4. Observe that the results are provided without proofs since (P ε ) 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.
From now on, for any µ > 0, the class of all families of nonempty subsets D = From now on, we assume a condition to simplify the exposition and the form of the limit problem (P 0 ), namely g 1 (0) = 1. (12) To prove the existence of a pullback absorbing family, we assume that there exists Remark 1. From the continuity of g 1 and (12) it is immediate to deduce that there existsε ∈ (0, 1] such thatμ < 2g 1 (ε)λ 1 m for all ε ∈ [0,ε]. Indeed, this last condition will be used in the sequel to construct the absorbing families in suitable universes D L 2 µε . Furthermore, it is consistent with the final goal of studying the limit of problems (P ε ) when ε goes to 0. In what follows the parameter µ ε is taken in [μ, 2g 1 (ε)λ 1 m) since (13) also holds with the weight eμ s replaced by e µεs and D L 2 µ ⊂ D L 2 µε , so a larger class of objects will be attracted.
Then, it holds the following result (cf. [6,Proposition 4] for a similar proof).
The proof of the following result is very close to that of [6, Proposition 5] under minor modifications.
Then, under the previous assumptions it fulfils that u ∈ L 2 (τ, T ; L 2 (Ω)). Therefore, it satisfies that u ∈ C([τ, T ]; H 1 0 (Ω)) and it holds Thanks to Theorem 2, the restriction of U ε to R 2 d × H 1 0 (Ω) defines a strict multivalued process into H 1 0 (Ω). Since no confusion arises, we will not modify the notation and continue denoting this process by U ε . Now to prove that the multi-valued process U ε is upper-semicontinuous with closed values in H 1 0 (Ω) for any ε fixed, we first provided the following auxiliary result.
On the other hand, using that {u n } n≥n2 is bounded in C([t − 2, t]; H 1 0 (Ω)), we have that for any sequence {s n } ⊂ [t − 2, t] with s n → s * , where (34) has been used to identify the weak limit. If we prove u n → u strongly in C([t − 1, t]; H 1 0 (Ω)), in particular, we will deduce that the sequence {u n (t)} is relatively compact in H 1 0 (Ω). To that end, we argue by contradiction. We suppose that there exist ε > 0, a sequence {t n } ⊂ [t − 1, t], without loss of generality converging to some t * , with From (35), it holds It is not difficult to prove, making use of the Galerkin approximations, that Then, we define the following continuous functions on [t − 2, t] Observe that thanks to (38), all the functions J n are non-increasing on the interval [t − 2, t]. In addition, taking into account the definition of J n and (33), it holds J n (s) → J(s) a.e. s ∈ (t − 2, t).
Since all the functions J n are non-increasing, for all n ≥ n( ) Then, lim sup n→∞ J n (t n ) ≤ J(t * ). Thus, it satisfies that lim sup n→∞ u n (t n ) ≤ u(t * ) which, together with (35), allow us to prove that {u n (t n )} converges to u(t * ) strongly in H 1 0 (Ω), in contradiction with (37). Therefore, (36) holds.
(43) Furthermore, for the sake of simplicity in the formulation of the limit problem, we also assume thatg Remark 3. (i) Under the new assumption (43), problem (P 0 ) becomes autonomous.
Since the above results also hold for (P 0 ), its associated pullback attractors are in fact the global compact attractor A 0 in L 2 (Ω) for the multi-valued semiflow S, where S(t − τ ) = U 0 (t, τ ). Namely, it can be seen as a pullback attractor for the universes D L 2 F and D L 2 µ0 with µ 0 = 2λ 1 m (cf. Proposition 2). Indeed, A 0 holds for all t ∈ R. . Namely, Finally, the upper semicontinuous behaviour of the pullback attractors A ε in H 1 -norm as ε goes to 0 for all t ∈ R is analysed. As done in [6], to prove these properties we first establish the following continuity (in ε) result of solutions to (P ε ) toward solutions of the limit problem and consider sequences {ε n } with lim n ε n = 0 and {u n τ } ⊂ L 2 (Ω) such that u n τ u τ weakly in L 2 (Ω). Then, there exist a subsequence of {u n τ } (relabeled the same), a sequence {u εn }, with u εn ∈ Φ εn (τ, u n τ ), and u 0 ∈ Φ 0 (τ, u τ ) such that u εn (t) → u 0 (t) strongly in H 1 0 (Ω) for all t > τ.
On the other hand, again from the energy equality (18) for the Galerkin approximations for (P εn,u εn ), applying (1) and (5) and passing to the limit, we have for Now, we define the following continuous functions on [τ + δ, T ] Observe that from (49) we deduce that all the functions J n are non-increasing on [τ + δ, T ]. Furthermore, since u εn (t) → u 0 (t) strongly in H 1 0 (Ω) a.e. t ∈ (τ + δ, T ), J is continuous in [τ + δ, T ] and all J n are non-increasing in [τ + δ, T ], it holds From this, we deduce Taking this into account, together with (48), (45) holds.
In order to prove the upper semicontinuous behaviour of attractors in H 1 0 (Ω) we introduce a last condition relating some terms involved in the formula for R ε L 2 when ε goes to 0, namely we assume that lim sup ε→0 (g 0 (ε)) 2 where µ ε are chosen in [μ, 2g 1 (ε)λ 1 m). Observe that in [6] we did not specify how to choose µ ε . Actually we just said that they could be taken equal to µ ε0 . In this paper, condition (50) provides how close to zero the amount 2g 1 (ε)m − λ −1 1 µ ε can be such that the whole fraction in (50) is O(1). This fact will be essential in the proof of our main result.

Remark 4. (i)
The results concerning weak solutions and L 2 -attractors only make use of f continuous 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 f just continuous. To that end, just simply considering mollifiers ρ δ , which implies that f δ = ρ δ * f fulfils (5), and compactness arguments.