Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms

Long-time behaviour of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms Tom´as Caraballo, Marta Herrera-Cobos, and Pedro Mar´ın-Rubio Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail addresses: [email protected], mh[email protected], [email protected] Dedicated to Enzo Mitidieri on occasion of his Sixtieth Birthday Abstract This paper is devoted to study the asymptotic behaviour of a time-dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth. Namely, we extend some previous results from the literature, obtaining existence, uniqueness, and continuity results, analyzing the stationary problem and decay of the solutions of the evolutionary problem, and finally, under more general assumptions, ensuring the existence of pullback attractors for the associated dynamical system in both L2and H1norms. Relationship among these objects are established using regularizing properties of the equation. 2010 MSC: 35B40, 35B41, 35B65, 35K55, 35Q92, 37L30. Key words and phrases: nonlocal diffusion, nonlocal terms, pullback attractors, regularity of attractors. 1 Introduction In recent decades, nonlocal problems have arisen in modeling with great interest by its usefulness in real applications (e.g. cf. [18, 4, 20, 30, 3]). For instance, in Biology, the evolution of some species might be better represented by a nonlocal equation than within the corresponding local simplification. Namely, as example of a spatial nonlocal feature, the migration of population of bacterias in a container depends on the global population in a certain subdomain (e.g. cf. [10, 11]). Of course, the disadvantage is that sometimes it is very complicated to deal with the nonlocal operators and terms since they are more involved. In particular, much attention has been paid to the parabolic PDE with nonlocal diffusion du dt −a(l(u))∆u=f, where the function aappearing in the diffusion coefficient satisfies suitable assumptions (detailed below) and depends on a nonlocal functional applied to the solution u. For instance, see [24, 9] for a detailed introduction on different applications, and for a very refined analysis of why imposing natural conditions of non-degeneracy of a(controlled from above and below) in order to avoid extinction and only existence of solutions in finite-time intervals. This problem is far away of being a trivial perturbation of a heat equation, since the nonlinear diffusion makes that the Lyapunov structure is, in general, lost, and manipulations that can be done in the local case are not valid here (e.g. roughly speaking, one cannot gain some information on a priori estimates by multiplying by ut). 1 Despite the above obstacles, the qualitative analysis of the evolution of solutions to the problem can be related to interesting questions as periodicity, or asymptotic-in-time issues –permanence vs extinction phenomena, stability–, etcetera, with meaningful interpretations, but more involved technical difficulties, which can only be solved in special cases. To be more precise, the results by Chipot and his collaborators (cited below) are referred to the above equation with findependent of u. There, besides existence and uniqueness results by different ways, the evolution for long times of the solutions is established under additional assumptions and particular suitably ordered initial conditions w.r.t. stationary solutions (e.g. see [10, 12]) or [13] for a mixed formulation in the boundary. As said before, a kind of Lyapunov structure were available only in very special situations (see [12, 14, 15]). Another modification of the above equation, with the nonlocal term in another position rather than in the diffusion, was also treated in [8], with similar conclusions (see also [7] for a model that includes as particular cases those cited above). In [1, 2] the authors analyze the case in which fis still independent of u, and the nonlocal operator is not acting globally in the whole domain but in the part of it contained in a ball centered on each position point. Radial solutions, bifurcation analysis, branch of solutions and their stability are studied. For fdepending on the unknown uin a semilinear form, the situation is more involved. In [29], by fixed point techniques, it is proved the existence and uniqueness of weak solution of a semilinear problem with nonlocal diffusion where the domain Ω has smooth boundary, and the term f(depending on u) is a Lipschitz continuous function. In addition, the existence and uniqueness of periodic solutions is also analyzed. Concerning the long-time behaviour of solutions, it is also worth considering the information that can be obtained by the theory of attractors, under less restrictive conditions on the problem. When fis independent of the solution and of time, this was also addressed by Andami [1] (see also some previous results on this direction by Lovat [24]). However, in this context of attractors, there are more choices to deal with, rather than the compact global attractor for an autonomous dynamical system. Indeed, after including time-dependent terms, which allows to model more complex situations, there are several different approaches from the point of view of non-autonomous dynamical systems, as uniform attractors, skew-product flows, and pullback attractors (e.g. cf. [22]; also related to random dynamical systems, cf. [17]). All of them are valid to analyze different features of the evolution of a non-autonomous dynamical system. We choose that of pullback attractors since it 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”). In this last approach of pullback attractors, many new results have appeared over the last years, allowing, as in the random case, to deal with not only fixed bounded sets, but developing the concept of attraction of a class of families (specifically, a universe D) parameterized in time, which usually appears in applications and use to be defined in term of a tempered condition (e.g. cf. [16, 5, 6]). Relationships between both type of attractors have already been established, and under suitable assumptions, they may actually coincide (cf. [27]). The goal of this paper is to study the existence and regularity of pullback attractors for dynamical systems generated by solutions of a parabolic PDE as above but of semilinear type, with nonlocal diffusion, and additional non-autonomous terms, extending the previously cited results. As far as we know, in the previous literature there is no study on the existence of the pullback attractor in L2(Ω) nor H1 0(Ω) for the associated dynamical system, as we will consider in this paper. As a first step towards this general aim, here we will be concerned with the sublinear case, to avoid additional difficulties that appear and which will be analyzed elsewhere in future. The content of the paper is as follows. In Section 2 we state the problem and some definitions of solutions and energy equalities that will be used later, and establish some existence results by compactness arguments. Section 3 is devoted to analyze the existence of stationary solution (combining 2 fixed point and compactness arguments), and uniqueness, under additional assumptions. We conclude this paragraph studying the global exponential stability of this steady state, i.e. exponential decay of the solutions to the evolution problem toward this unique stationary solution. In Section 4 we briefly recall some abstract results on pullback attractors that will be used in the sequel. Then, in Section 5 we establish the existence of several pullback attractors in L2(Ω). The proof is based on an energy method which relies on the continuity of solutions. In addition, we establish some relationships between these pullback attractors. Finally, in Section 6 we use a similar argument to ensure the existence of pullback attractors in H1 0(Ω).Moreover, under suitable assumptions and taking into account the regularizing effect of the equation, we can establish relationships between these families of pullback attractors. 2 Statement of the problem and existence results Consider the following problem for a nonlocal and nonlinear parabolic equation:      du dt −a(l(u))∆u=f(u) + h(t) in Ω ×(τ, +∞), u= 0 on ∂Ω×(τ, +∞), u(x, τ) = uτ(x) in Ω, (1) where Ω ⊂RNis a bounded open set, τ∈R, a ∈C(R;R+) is locally Lipschitz, and there exist positive constants m, M such that 0< m ≤a(s)≤M∀s∈R,(2) l∈(L2(Ω))′,f∈C(R) and there exist constants η > 0,and Cf≥0,such that |f(s)| ≤ Cf(1 + |s|)∀s∈R,(3) (f(s)−f(r))(s−r)≤η(s−r)2∀s, r ∈R.(4) As usual, we will denote by (·,·) the inner product in L2(Ω) and by |·|its associated norm (since no confusion arises, this will also denote the Lebesgue measure of a subset of RN). By ((·,·)) we will denote the inner product in H1 0(Ω) given by the product in (L2(Ω))Nof the gradients, by k · k the associated norm, by h·,·i the duality product between H−1(Ω) and H1 0(Ω),and by k · k∗the norm in H−1(Ω).Identifying L2(Ω) with its dual, we have the usual chain of dense and compact embeddings H1 0(Ω) ⊂L2(Ω) ⊂H−1(Ω).Observe that, by the Riesz theorem, we can obtain ˜ l∈L2(Ω) with hl, ui(L2(Ω))′,L2(Ω) = (˜ l, u); here on, thanks to the identification (L2(Ω))′≡L2(Ω), we will make the abuse of notation of using linstead of ˜ l, but at the same time we keep the usual notation in the existing previous literature l(u) instead of (l, u) for the operator lacting on u. In what follows, we assume that uτ∈L2(Ω) and h∈L2 loc(R;H−1(Ω)). Definition 1. A weak solution to (1) is a function uthat belongs to L2(τ, T ;H1 0(Ω))∩L∞(τ, T ;L2(Ω)) for all T > τ, with u(τ) = uτ,such that d dt(u(t), v) + a(l(u(t)))((u(t), v)) = (f(u(t)), v) + hh(t), vi ∀v∈H1 0(Ω),(5) where the previous equation must be understood in the sense of D′(τ, +∞). Remark 2. If uis a weak solution to (1), then (2), (3), and (5) imply that u′∈L2(τ, T ;H−1(Ω)) for any T > τ, and therefore u∈C([τ, +∞); L2(Ω)).Hence the initial datum in (1) makes sense. Moreover, we have the following energy equality for all τ≤s≤t: |u(t)|2+ 2 Zt s a(l(u(r)))ku(r)k2dr =|u(s)|2+ 2 Zt s (f(u(r)), u(r))dr + 2 Zt s hh(r), u(r)idr. (6) 3 A notion of more regular solution is also suitable for the problem. Definition 3. A strong solution to (1) is a weak solution uwhich also satisfies that u∈L2(τ, T ;D(−∆)) ∩L∞(τ, T ;H1 0(Ω)) for all T > τ. Remark 4. If h∈L2 loc(R;L2(Ω)) and uis a strong solution to (1), then u′∈L2(τ, T ;L2(Ω)) for all T > τ, and, consequently, u∈C([τ, +∞); H1 0(Ω)). In addition, the following energy equality holds: ku(t)k2+ 2 Zt s a(l(u(r)))| − ∆u(r)|2dr =ku(s)k2+ 2 Zt s (f(u(r)) + h(r),−∆u(r))dr ∀τ≤s≤t. (7) Next result establishes the existence and uniqueness of weak and strong solution to (1), the regularizing effect of the equation, and the continuity of the solution in L2(Ω) with respect to initial data. Theorem 5. Assume that the function ais locally Lipschitz and satisfies (2), f∈C(R)fulfills (3) and (4), and consider h∈L2 loc(R;H−1(Ω)) and l∈L2(Ω) given. Then, for each uτ∈L2(Ω), the problem (1) possesses a unique weak solution, denoted by u(·) = u(·;τ, uτ). Moreover, this solution behaves continuously in L2(Ω) w.r.t. initial data. In addition, if h∈L2 loc(R;L2(Ω)), for every ε > 0and T > τ +ε, this solution usatisfies that u∈C((τ, T ], H1 0(Ω)) ∩L2(τ+ε, T ;D(−∆)).In fact, if the initial condition uτ∈H1 0(Ω), then u∈C([τ, T ], H1 0(Ω)) ∩L2(τ, T ;D(−∆)) for every T > τ, i.e. uis a strong solution. Proof. We split the proof into three steps. Step 1. Uniqueness of solution and continuity w.r.t. initial data. We will prove both assertions simultaneously since the same estimates are valid for both purposes. Suppose that u1and u2are two weak solutions to (1) corresponding to initial values u1τ, u2τ∈L2(Ω) respectively. From the energy equality, 1 2 d dt|u1(t)−u2(t)|2+a(l(u1(t)))ku1(t)−u2(t)k2 = [a(l(u2(t))) −a(l(u1(t)))]((u2(t), u1(t)−u2(t))) + (f(u1(t)) −f(u2(t)), u1(t)−u2(t)) a.e. t∈[τ, T ]. Since u1, u2∈C([τ, T ]; L2(Ω)),it holds that u1(t), u2(t)∈Sfor all t∈[τ, T ],where Sis a bounded set of L2(Ω).On the other hand, as l∈L2(Ω), there exists a value R > 0 such that {l(ui(t))}t∈[τ,T ]⊂[−R, R] for i= 1,2. Hence, using (2), (4), and the locally Lipschitz continuity of the function a, it yields 1 2 d dt|u1(t)−u2(t)|2+mku1(t)−u2(t)k2 ≤La(R)|l||u1(t)−u2(t)|ku2(t)kku1(t)−u2(t)k+η|u1(t)−u2(t)|2, where La(R) is the Lipschitz constant of the function ain [−R, R].Thanks to the Young inequality, d dt|u1(t)−u2(t)|2≤(2m)−1(La(R))2|l|2ku2(t)k2+ 2η|u1(t)−u2(t)|2. Therefore, |u1(t)−u2(t)|2≤exp Zt τ(2m)−1(La(R))2|l|2ku2(s)k2+ 2ηds|u1τ−u2τ|2, whence both statements, uniqueness (when u1τ=u2τ) and continuity w.r.t. initial data follow. 4 Step 2. Existence of weak solution. Assume that uτ∈L2(Ω) and h∈L2 loc(R;H−1(Ω)).By the spectral theory, we can consider {wj} ⊂ H1 0(Ω),a Hilbert basis of L2(Ω) formed by eigenfunctions of −∆ with zero Dirichlet boundary condition in Ω, with corresponding eigenvalues 0 < λ1≤λ2≤... For every integer n≥1, we denote by un(t;τ, uτ) = Pn j=1 ϕnj(t)wj, un(·) for short, the (Galerkin approximation) solution of    d dt(un(t), wj) + a(l(un(t)))((un(t), wj)) = (f(un(t)), wj) + hh(t), wji, t > τ (un(τ), wj) = (uτ, wj), j = 1,...,n. (8) Multiplying by ϕnj(t) in (8), summing from j= 1 to n, and using (2), we obtain d dt|un(t)|2+ 2mkun(t)k2≤2(f(un(t)), un(t)) + 2hh(t), un(t)ia.e. t > τ. (9) By the Young inequality and (3), d dt|un(t)|2+mkun(t)k2≤4C2 f|Ω| λ1m+4C2 f λ1m|un(t)|2+2 mkh(t)k2 ∗a.e. t > τ. (10) Hence, from the Gronwall lemma we deduce that {un}is bounded in L∞(τ, T ;L2(Ω))∩L2(τ, T ;H1 0(Ω)). As a consequence, {−a(l(un))∆un}is bounded in L2(τ, T ;H−1(Ω)),and using (3), {f(un)}is bounded in L2(τ, T ;L2(Ω)).Then, it is not difficult to prove that {u′ n}is bounded in L2(τ, T ;H−1(Ω)). Therefore, from compactness arguments and the Aubin-Lions lemma, there exist a subsequence of {un} (relabelled the same) and u∈L∞(τ, T ;L2(Ω))∩L2(τ, T ;H1 0(Ω)) with u′∈L2(τ, T ;H−1(Ω)),such that un ∗ ⇀ u weakly-star in L∞(τ, T ;L2(Ω)), un⇀ u weakly in L2(τ, T ;H1 0(Ω)), u′ n⇀ u′weakly in L2(τ, T ;H−1(Ω)), f(un)⇀ ξ1weakly in L2(τ, T ;L2(Ω)), a(l(un))un⇀ ξ2weakly in L2(τ, T ;H1 0(Ω)), un→ustrongly in L2(τ, T ;L2(Ω)), un(x, t)→u(x, t) a.e. (x, t)∈Ω×(τ, T ), un(t)→u(t) strongly in L2(Ω) a.e. t∈(τ, T ) for all T > τ. Then, from [23, Lemme 1.3, p. 12], it is not difficult to prove that ξ1=f(u) and ξ2=a(l(u))u. Thus, we can pass to the limit in (8), and thanks to the fact that ∪n∈Nspan[w1,...,wn] is dense in H1 0(Ω),we conclude that uis a weak solution. Step 3. Regularizing effect and strong solution. Integrating in (10) between τand T, in particular mZT τ kun(s)k2ds ≤ |uτ|2+4C2 f|Ω| λ1m(T−τ) + 4C2 f λ1mZT τ |un(s)|2ds +2 mZT τ kh(s)k2 ∗ds. (11) Assuming now that h∈L2 loc(R;L2(Ω)),multiplying by λjϕnj(t) in (8), summing from j= 1 to n, and using (2), (3), and the Young and Poincar´e inequalities, we obtain d dtkun(t)k2+m| − ∆un(t)|2≤4C2 f|Ω| m+4C2 f λ1mkun(t)k2+2 m|h(t)|2a.e. t≥τ. 5 Integrating the previous expression between sand t, with τ≤s≤t≤T, kun(t)k2+mZt s | − ∆un(r)|2dr ≤4C2 f|Ω|(T−τ) m+4C2 f λ1mZt s kun(r)k2dr +2 mZt s |h(r)|2dr +kun(s)k2.(12) In particular, integrating again with respect to sbetween τand t, it holds (t−τ)kun(t)k2≤4C2 f|Ω|(T−τ)2 m+ 4C2 f(T−τ) λ1m+ 1!ZT τ kun(r)k2dr +2(T−τ) mZT τ |h(r)|2dr. Therefore, for all t∈[ε+τ, T ] with ε∈(0, T −τ), it holds kun(t)k2≤4C2 f|Ω|(T−τ)2 εm + 4C2 f(T−τ) + λ1m ελ1m!ZT τ kun(r)k2dr +2(T−τ) εm ZT τ |h(r)|2dr. From this and (11) we deduce that the sequence {un}is bounded in L∞(ε+τ, T ;H1 0(Ω)). On the other hand, taking s=εand t=Tin (12), and using the previous estimates, we obtain that {un}is bounded in L2(τ+ε, T ;D(−∆)).As a consequence, {u′ n}is bounded in L2(τ+ ε, T ;L2(Ω)),and thanks to the uniqueness of the weak solution, it holds that {un}converges to uweakly in L2(τ+ε, T ;D(−∆)) and {u′ n}converges to u′weakly in L2(τ+ε, T ;L2(Ω)).Thus, u∈L2(τ+ε, T ;D(−∆)) ∩C((τ, T ]; H1 0(Ω)). The case in which the initial datum uτbelongs to H1 0(Ω) allows to simplify the above estimates in a standard way, and the solution becomes actually strong. 3 Stationary solutions and their stability In this section we are interested in proving that problem (1) admits stationary solutions under some extra assumptions. In order to do this, we will use a corollary of the Brouwer fixed point theorem. Also, under additional suitable requirements, we will prove that in fact the stationary solution is unique and is globally asymptotically exponentially stable. As we aim now to deal with stationary (steady-state) solutions for the problem, we assume that h is time-independent, i.e. h∈H−1(Ω).We also assume in this paragraph that the functions aand f are globally Lipschitz, with respective Lipschitz constants La, Lf≥0. Once that we consider the evolutionary problem (1) under the above assumptions, by a stationary solution to the problem, we mean an element u∗∈H1 0(Ω) such that a(l(u∗))((u∗, v)) = (f(u∗), v) + hh, vi ∀v∈H1 0(Ω).(13) We then can prove the following result concerning existence, uniqueness, and regularity of stationary solutions to the problem (the idea of the proof is close to that in [25]). Theorem 6. Assume that the functions aand fare globally Lipschitz functions, with Lipschitz constants Laand Lfrespectively, (2) is satisfied, h∈H−1(Ω), l ∈L2(Ω),and m > λ−1 1Lf.Then: 1. Problem (13) admits at least one solution. Moreover, any solution u∗to (13) satisfies ku∗k ≤ Υ := λ−1/2 1|Ω|1/2|f(0)|+khk∗ m−λ−1 1Lf .(14) In addition, if h∈L2(Ω), then the solutions given above belong in fact to D(−∆). 6 2. Besides, if we assume λ−1/2 1|l|LaΥ< m −λ−1 1Lf,(15) problem (13) possesses a unique solution. Proof. We split the proof of the two statements. Step 1. Existence. As in the proof of Theorem 5, we choose the Hilbert basis {wj}of L2(Ω), formed by the eigenvectors (with corresponding eigenvalues {λj}) of the operator −∆ in Ω with zero Dirichlet boundary condition. For each n≥1,let us denote Vn= span[w1,...,wn],with the inner product ((·,·)) and norm k · k.Then, for all n≥1,we define the operators Rn:Vn→Vnas follows, ((Rnu, v)) = h−a(l(u))∆u, vi − (f(u), v)− hh, vi ∀u, v ∈Vn. Since the right hand side is a continuous linear map from Vnto R,the Riesz Theorem ensures that each Rnu∈Vnis well defined. Now, let us check that Rnis continuous. Indeed, observe that ((Rnu−Rn˜u, v)) = h−a(l(u))∆u+a(l(˜u))∆˜u−f(u) + f(˜u), vi =h−a(l(u))∆(u−˜u) + (a(l(˜u)) −a(l(u)))∆˜u, vi+ (f(˜u)−f(u), v) ≤(Mku−˜uk+La|l|λ−1/2 1k˜u−ukk˜uk+Lfλ−1 1k˜u−uk)kvk for all u, ˜u, v ∈Vn,thanks to the assumptions on the functions a, f, and l. Therefore, kRnu−Rn˜uk ≤ (M+λ−1/2 1|l|Lak˜uk+λ−1 1Lf)k˜u−uk. for all u, ˜u∈Vn.This proves that Rn:Vn→Vnis continuous. On the other hand, ((Rnu, u)) = h−a(l(u))∆u, ui − (f(u), u)±(f(0), u)− hh, ui ≥mkuk2−λ−1 1Lfkuk2− |f(0)|λ−1/2 1|Ω|1/2kuk − khk∗kuk(16) for all u∈Vn. Hence, we obtain ((Rnu, u)) ≥0 for all u∈Vnsuch that kuk= Υ,the value given in (14). Consequently, by a corollary of the Brouwer fixed point theorem (see [23, Lemme 4.3, p. 53]), for each n≥1 there exists un∈Vnsuch that Rn(un) = 0,with kunk ≤ Υ. Therefore, we deduce that h−a(l(un))∆un, vi= (f(un), v) + hh, vi ∀v∈Vn. Since the sequence {un}is bounded in H1 0(Ω), the compact embedding of H1 0(Ω) in L2(Ω) allows us to extract a subsequence {un′} ⊂ {un}that converges weakly in H1 0(Ω) and strongly in L2(Ω) to an element u∗∈H1 0(Ω),which also verifies (14). Passing to the limit in the above equality, and taking into account the assumptions on the functions a,f, and l, we obtain that u∗is a solution to (13). The a priori estimate (14). It is deduced in a similar way as we argued in (16). Regularity. Assume now that h∈L2(Ω),and consider any stationary solution u∗of the problem (13). We must prove that u∗∈D(−∆).Obviously, u∗is solution to the evolutionary problem (1) with initial datum u(τ) = u∗.Therefore, by the regularity result in Theorem 5, u∗∈D(−∆). Step 2. Uniqueness. Let us suppose that u1and u2are two solutions to (13). Then, h−a(l(u1))∆u1+a(l(u2))∆u2, vi= (f(u1)−f(u2), v) for all v∈H1 0(Ω). Introducing ±a(l(u1))∆u2and taking v=u1−u2as test function, we obtain mku1−u2k2≤(λ−1/2 1|l|Laku2k+λ−1 1Lf)ku1−u2k2. 7 If u16=u2,we can simplify the above expression, dropping the factor ku1−u2k2. But then, using the a priori estimate (14) for u2, we would arrive at the opposite inequality to that one in (15), what is a contradiction. Thus, it must be u1=u2, and the proof is concluded. Our second result in this section establishes the global asymptotic exponential stability of the unique stationary solution obtained previously. Theorem 7. Under the assumptions in Theorem 6, and if (15) holds, then, the following estimate holds for the difference of any solution to (1) and the unique solution u∗to (13): |u(t;τ, uτ)−u∗|2≤e−λ(t−τ)|uτ−u∗|2∀t≥τ, where λ= 2λ1(m−λ−1 1Lf−λ−1/2 1|l|LaΥ) >0. Proof. For short, denote by u(·) the weak solution of (1). Then, from the energy equality, 1 2 d dt|u(t)−u∗|2=ha(l(u(t)))∆u(t)−a(l(u∗))∆u∗+f(u(t)) −f(u∗), u(t)−u∗ia.e. t∈(τ, T ). Introducing ±a(l(u))∆u∗,and using (2), the Poincar´e inequality, and the fact that the functions aand fare globally Lipschitz, it holds 1 2 d dt|u(t)−u∗|2=−a(l(u(t)))ku(t)−u∗k2+h(a(l(u(t))) −a(l(u∗)))∆u∗+f(u(t)) −f(u∗), u(t)−u∗i ≤(−m+λ−1/2 1|l|Laku∗k+Lfλ−1 1)ku(t)−u∗k2. From (14), (15), and thanks once again to the Poincar´e inequality, we arrive at d dt|u(t)−u∗|2≤ −λ|u(t)−u∗|2a.e. t > τ, where λis given in the statement. The proof is therefore complete. 4 Abstract results on the theory of Pullback Attractors The aim of this section is to recall briefly the main results from the theory of pullback attractors which will be used in the sequel (e.g. cf. [5, 6, 19, 27]). Consider given a metric space (X, dX),and let us denote R2 d={(t, τ)∈R2:τ≤t}. Definition 8. (a) A process on X(also called a two-parameter semigroup) is a mapping R2 d×X∋(t, τ, x)7→ U(t, τ)x∈Xsuch that U(τ, τ)x=xfor any (τ, x)∈R×X, and U(t, s)(U(s, r)x) = U(t, r)xfor any r≤s≤tand all x∈X. (b) A process Uon Xis said to be 1. continuous if for any pair (t, τ)∈R2 d, the mapping U(t, τ) : X→Xis continuous; 2. strong-weak (also known as norm-to-weak) continuous if for any pair (t, τ)∈R2 d,the map U(t, τ)is continuous from Xwith the strong topology into Xwith the weak topology; 3. closed if for any pair (t, τ)∈R2 d, and any sequence {xn} ⊂ X, if xn→x∈Xand U(t, τ)xn→y∈X, then U(t, τ)x=y. 8 It is clear that every continuous process is strong-weak continuous, and every strong-weak continuous process is closed. Let us denote by P(X) the family of all nonempty subsets of Xand consider Da nonempty class of families parameterized in time b D={D(t) : t∈R} ⊂ P(X) and a family of nonempty sets b D0={D0(t) : t∈R} ⊂ P(X).Observe what we do not require any additional condition on these sets such as compactness or boundedness. Definition 9. A process Uon Xis said to be pullback D-asymptotically compact if for any t∈R, any b D∈ D,and any sequences {τn} ⊂ (−∞, t]and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D(τn)for all n, the sequence {U(t, τn)xn}is relatively compact in X. The family b D0={D0(t) : t∈R} ⊂ P(X)is pullback D-absorbing for the process Uon Xif for any t∈Rand any b D∈ D, there exists τ0(b D, t)< t such that U(t, τ)D(τ)⊂D0(t)for all τ≤τ0(b D, t). Observe that in the above definition b D0does not necessarily belong to the class D. Definition 10. The family AD={AD(t) : t∈R} ⊂ P(X)is said to be the minimal pullback D-attractor for the process Uif it satisfies the following properties: 1. for any t∈R,the set AD(t)is a nonempty compact subset of X; 2. ADis pullback D-attracting, i.e. limτ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D, t ∈R, where distX(·,·)denotes the Hausdorff semi-distance in Xbetween two subsets of X; 3. ADis invariant, i.e. U(t, τ)AD(τ) = AD(t)for all τ≤t; 4. ADis minimal in the sense that if b C={C(t) : t∈R} ⊂ P(X)is a family of closed sets which is pullback D-attracting, then AD(t)⊂C(t)for all t∈R. The following result ensures the existence of a minimal pullback attractor (cf. [19, Theorem 3.11]). Theorem 11. Consider a closed process U:R2 d×X→X, a universe Din P(X),and a family b D0={D0(t) : t∈R} ⊂ P(X)which is pullback D-absorbing for Uand assume also that Uis pullback b D0-asymptotically compact. Then, the family AD={AD(t) : t∈R}defined by AD(t) = Sb D∈D Λ( b D, t) X for t∈R,where Λ( b D, t) = Ts≤tSτ≤sU(t, τ)D(τ)Xand {...}Xdenotes the closure in X, is the minimal pullback D-attractor for the process U. Moreover, if b D0∈ D, then AD(t)⊂D0(t)Xfor all t∈R. Let us 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) = D:t∈R}with Da fixed nonempty bounded subset of X. In the particular case of the universe DX F,the corresponding minimal pullback DX F-attractor for the process Uis the pullback attractor defined by Crauel, Debussche, and Flandoli (see [17]). Then, we have the following relationships between pullback attractors (see [27]). Corollary 12. Under the assumptions of Theorem 11, if DX F⊂ D, it holds that both attractors, ADX F and AD, exist and ADX F(t)⊂ AD(t)∀t∈R. Remark 13. Under the assumptions of the previous corollary, if for some T∈Rthe set St≤TD0(t) is a bounded subset of X, then ADX F(t) = AD(t)∀t≤T. The following result allows to compare two attractors for a process under appropriate assumptions (see [19]). 9 Proof. Consider fixed (τ, t)∈R2 d. Let {uτn}be a sequence that converges to uτstrongly in H1 0(Ω). On the one hand, by Proposition 15, it holds that the map U(t, τ) is continuous from L2(Ω) into L2(Ω). Therefore, U(t, τ)uτn→U(t, τ)uτstrongly in L2(Ω). On the other hand, using (7), (2), (3), and the H¨older and Young inequalities, it holds kU(t, τ)uτnk2≤ kuτnk2+2C2 f mZt τ (|Ω|+|U(s, τ)uτn|2)ds+1 mZt τ |h(s)|2ds. From (6), and the Gronwall lemma, in particular, a uniform estimate for {U(·, τ)uτn}in L2(τ, t;L2(Ω)) follows easily. Hence, the sequence {U(t, τ)uτn}is bounded in H1 0(Ω).Then, by the uniqueness of the limit, it holds U(t, τ)uτn⇀ U(t, τ)uτweakly in H1 0(Ω). Next result, which is analogous to Lemma 20, establishes some uniform estimates of the solutions in more regular norms in a finite-time interval up to time twhen the initial datum is shifted pullback far enough. This will be useful to prove the pullback asymptotic compactness in H1 0(Ω). In order to simplify the statement, let us firstly introduce the following two quantities: ρext 1(t) = 1 + 2β|Ω| µ+e−µ(t−3) 2(m−αλ−1 1)−µλ−1 1Zt −∞ eµθkh(θ)k2 ∗dθ, ρext 2(t) = 1 m−αλ−1 12β|Ω|+ρext 1(t) + 1 m−αλ−1 1 max r∈[t−2,t]Zr r−1 kh(θ)k2 ∗dθ. (29) [The upper script ext means that these expressions are estimates, close to those in Lemma 20 involving ρ1and ρ2, but in an extended interval, as will be indicated in the proof below.] Lemma 25. Assume that the function ais locally Lipschitz and satisfies (2), l∈L2(Ω),f∈C(R) satisfies (3), (4), and (17), and h∈L2 loc(R;L2(Ω)) satisfies (18) for some µ∈(0,2(λ1m−α)). Then, for any t∈Rand b D∈ DL2 µ,there exists τ2(b D, t)< t −3,such that for any τ≤τ2(b D, t)and any uτ∈D(τ),it holds                    ku(r;τ, uτ)k2≤eρ1(t)∀r∈[t−2, t], Zr r−1 | − ∆u(θ;τ, uτ)|2dθ ≤eρ2(t)∀r∈[t−1, t], Zr r−1 |u′(θ;τ, uτ)|2dθ ≤eρ3(t)∀r∈[t−1, t], (30) where, taking into account {ρext i}i=1,2from (29), the terms {eρi}i=1,2,3are given by eρ1(t) = 4C2 f|Ω| m+ 1 + 4C2 f λ1m!ρext 2(t) + 2 mmax r∈[t−2,t]Zr r−1 |h(θ)|2dθ, eρ2(t) = 4C2 f|Ω| m2+1 meρ1(t) + 4C2 f λ1m2ρext 2(t) + 2 m2max r∈[t−1,t]Zr r−1 |h(θ)|2dθ, eρ3(t) = 3M2eρ2(t) + 6C2 f|Ω|+6C2 f λ1 ρext 2(t) + 3 max r∈[t−1,t]Zr r−1 |h(θ)|2dθ. 16 Proof. Let us firstly observe that, analogously as we argued in Lemma 20, we may obtain uniform estimates for solutions in a longer time-interval (useful for our purposes). Namely, there exists τ2(b D, t)< t −3,such that for any τ≤τ2(b D, t) and any uτ∈D(τ),it holds |u(r;τ, uτ)|2≤ρext 1(t)∀r∈[t−3, t], Zr r−1 ku(θ;τ, uτ)k2dθ ≤ρext 2(t)∀r∈[t−2, t], where {ρext i}i=1,2are given in (29). Actually, these estimates also hold for the Galerkin approximations un(·;τ, uτ). Taking this into account, we will prove the lemma by obtaining the inequalities for the Galerkin approximations and then passing to the limit. Now, multiplying by λjϕnj(θ) in (8), summing from j= 1 to nand using (2), (3), and the Young inequality, we obtain d dθ kun(θ)k2+m| − ∆un(θ)|2≤4C2 f|Ω| m+4C2 f λ1mkun(θ)k2+2 m|h(θ)|2a.e. θ > τ (31) Integrating between rand swith τ≤r−1≤s≤r, we obtain in particular kun(r)k2≤ kun(s)k2+4C2 f|Ω| m+4C2 f λ1mZr r−1 kun(θ)k2dθ +2 mZr r−1 |h(θ)|2dθ. Integrating the last inequality w.r.t. son [r−1, r], it holds kun(r)k2≤ 1 + 4C2 f λ1m!Zr r−1 kun(s)k2ds +4C2 f|Ω| m+2 mZr r−1 |h(θ)|2dθ, for all τ≤r−1. Therefore, from the estimate on the solutions by ρext 2given above, one deduces that kun(r;τ, uτ)k2≤eρ1(t)∀r∈[t−2, t], τ ≤τ2(b D, t), uτ∈D(τ), where eρ1(t) is given in the statement. Now, integrating between r−1 and rin (31), we obtain in particular Zr r−1 | − ∆un(θ)|2dθ ≤1 mkun(r−1)k2+4C2 f|Ω| m2+4C2 f λ1m2Zr r−1 kun(θ)k2dθ +2 m2Zr r−1 |h(θ)|2dθ for all τ≤r−1. Therefore, for any n≥1, Zr r−1 | − ∆un(θ)|2dθ ≤eρ2(t)∀r∈[t−1, t], τ ≤τ2(b D, t), uτ∈D(τ), where eρ2(t) is given in the statement. On the other hand, Zr r−1 |u′ n(θ)|2dθ ≤3Zr r−1 a(l(un(θ)))2| − ∆un(θ)|2dθ + 3 Zr r−1 |f(un(θ))|2dθ + 3 Zr r−1 |h(θ)|2dθ, for all τ≤r−1.Therefore, from (2), (3), and the above estimates, for any n≥1,it yields Zr r−1 |u′ n(θ)|2dθ ≤eρ3(t)∀r∈[t−1, t], τ ≤τ2(b D, t), uτ∈D(τ), 17 where eρ3(t) is given in the statement. Using that u∈C([t−2, t]; H1 0(Ω)) and the facts that {un}converges to u(·;τ, uτ) weakly-star in L∞(t−2, t;H1 0(Ω)) and weakly in L2(r−1, r;D(−∆)) for all r∈[t−1, t], and {u′ n}converges to u′(·;τ, uτ) weakly in L2(r−1, r;L2(Ω)) for all r∈[t−1, t],from the above estimates, passing to the limit we conclude (30). Now, we introduce additional universes, that involve more regularity. Definition 26. For each µ > 0, we will denote by DL2,H1 0 µthe class of all families of nonempty subsets b DH1 0={D(t)∩H1 0(Ω) : t∈R}, where b D={D(t) : t∈R} ∈ DL2 µ. Remark 27. Observe that DH1 0 F⊂ DL2,H1 0 µand DL2,H1 0 µis inclusion-closed. As a direct consequence of the regularizing effect of the equation when h∈L2 loc(R;L2(Ω)) (cf. Theorem 5) and the existence of a family pullback DL2 µ-absorbing (cf. Proposition 19), the existence of an absorbing family in this universe DL2,H1 0 µalso holds. Proposition 28. Under the assumptions of Lemma 25, the family b D0,H1 0={B(0, R1/2 L2(t)) ∩H1 0(Ω) : t∈R} belongs to DL2,H1 0 µand for any t∈Rand any b D∈ DL2 µ,there exists τ3(b D, t)< t such that U(t, τ)D(τ)⊂D0,H1 0(t)∀τ≤τ3(b D, t). In particular, the family b D0,H1 0is pullback DL2,H1 0 µ-absorbing for the process U:R2 d×H1 0(Ω) →H1 0(Ω). Now, we apply an energy method analogous to the one we used in the previous section. This will lead to pullback asymptotic compactness of Uin H1 0(Ω) for the universe DL2,H1 0 µ. Proposition 29. Under the assumptions of Lemma 25, the process U:R2 d×H1 0(Ω) →H1 0(Ω) is pullback DL2,H1 0 µ-asymptotically compact. Proof. The proof is similar to that of Lemma 21, and we omit it for the sake of brevity. Let us just indicate the main key in this more regular setting: in this case we consider the energy equality (7) and the continuous and non-increasing functions Jn(s) = kun(s)k2−4C2 f|Ω| ms−4C2 f mZs t−2 |un(r)|2dr −2 mZs t−2 |h(r)|2dr, J(s) = ku(s)k2−4C2 f|Ω| ms−4C2 f mZs t−2 |u(r)|2dr −2 mZs t−2 |h(r)|2dr. As a consequence of the above results, we obtain the existence of minimal pullback attractors for the process U:R2 d×H1 0(Ω) →H1 0(Ω). Theorem 30. Suppose that the function ais locally Lipschitz and satisfies (2), f∈C(R)fulfills (3), (4), and (17), l∈L2(Ω), and h∈L2 loc(R, L2(Ω)) verifies (18) for some µ∈(0,2(λ1m−α)). Then, there exist the minimal pullback DH1 0 F-attractor ADH1 0 F ={ADH1 0 F (t) : t∈R}and the minimal pullback 18 DL2,H1 0 µ-attractor ADL2,H1 0 µ ={ADL2,H1 0 µ (t) : t∈R},for the process U:R2 d×H1 0(Ω) →H1 0(Ω), and it holds ADH1 0 F (t)⊂ ADL2 F (t)⊂ ADL2 µ(t) = ADL2,H1 0 µ (t)∀t∈R,(32) where ADL2 F and ADL2 F are respectively the minimal pullback DL2 F-attractor and the minimal pullback DL2 µ-attractor for the process U:R2 d×L2(Ω) →L2(Ω), whose existence is guaranteed by Theorem 22. In particular, it holds the following pullback attraction result lim τ→−∞ distH1 0(U(t, τ)D(τ),ADL2 µ(t)) = 0 ∀t∈R∀b D∈ DL2 µ.(33) Finally, if moreover hsatisfies not only (28) but sup s≤0e−µs Zs −∞ eµr|h(r)|2dr<+∞,(34) then all attractors in (32) coincide, i.e. ADH1 0 F(t)=ADL2 F (t) = ADL2 µ(t) = ADL2,H1 0 µ (t)∀t∈R,(35) and again, a result of pullback attraction holds, in this case for any bounded subset Bof L2(Ω) : lim τ→−∞ distH1 0(U(t, τ)B, ADL2 F (t)) = 0 ∀t∈R.(36) Proof. The existence of ADH1 0 F and ADL2,H1 0 µ is a consequence of Corollary 12, since Uis strong-weak continuous in H1 0(Ω) (cf. Proposition 24), the relation between the universes (cf. Remark 27), and we have the existence of an absorbing family in DL2,H1 0 µ(cf. Proposition 28) and asymptotic compactness of this universe in the H1norm (cf. Proposition 29). The chain of inclusions (32) follows from Corollary 12 and Theorem 14. Actually, the equality statement is due to the second part of Theorem 14, by using Proposition 28. Then, (33) is straightforward. If moreover hsatisfies (28), we already proved in Theorem 22 the equality ADL2 F (t) = ADL2 µ(t) for all t∈R.Now, in order to obtain (35), we assume (34), which is a requirement stronger than (28). Therefore, the equality ADH1 0 F (t) = ADL2 F (t) is again a consequence of Theorem 14. Indeed, the solutions are coming into a bounded subset of H1 0(Ω) due to the first estimate in Lemma 25 by eρ1(t) [recall that, analogously as in Remark 23 (i), here (34) is equivalent to sups≤0Rs s−1|h(r)|2dr < +∞]. Then, (36) obviously holds. Remark 31. (i) As a complement to Remark 23, under the assumptions of Theorem 30, we can improve the above result, involving new universes and attractors. Namely, for any σ∈(µ, 2(λ1m−α)) there exists the corresponding minimal pullback DL2,H1 0 σ-attractor, ADL2,H1 0 σ ,and it holds that ADL2 σ(t) = ADL2,H1 0 σ (t)for all t∈R.Moreover, if hsatisfies (34), then, after (35), ADL2,H1 0 σ coincides with the family ADH1 0 F for any σ∈(µ, 2(λ1m−α)). (ii) Under additional conditions, we may restrict ourselves to study the problem in the positive cone of L2(Ω),that is, C+(L2(Ω)) = {g∈L2(Ω) : g≥0a.e. Ω}, and redefining suitably new classes of (tempered and non-tempered) families. Namely, assuming that his a positive function and f∈C(R;R+), a Maximum Principle holds, and Uis well-defined from into C+(L2(Ω)), which is important if one is dealing with a biological model. Then, all the results from Sections 5 and 6 can be obtained again analogously, by rearranging the assumptions within this setting. 19 Acknowledgments While concluding this work, we received the very sad news that Professor Silvano B. de Menezes passed away during a stay at Universidad de Sevilla, visiting our department. We feel extremely sorrow for this news, for him, his family and friends, and the big community of mathematicians that knew him. The authors would like to thank the referees for a careful reading and comments on a previous version of the manuscript. This work has been partially supported by Ministerio de Econom´ıa y Competitividad (Spain), grant MTM2011-22411, Junta de Andaluc´ıa grant P12-FQM-1492. 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