Regularity Results and Exponential Growth for Pullback Attractors of a Non-Autonomous Reaction-Diffusion Model with Dynamical Boundary Conditions

Regularity results and exponential growth for pullback attractors of a non-autonomous reaction-diffusion model with dynamical boundary conditions Mar´ıa Anguiano, Pedro Mar´ın-Rubio, Jos´e Real∗ Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla (Spain) e-mails: [email protected] [email protected] May 7, 2014 Abstract In this paper, we prove some regularity results for pullback attractors of a nonautonomous reaction-diffusion model with dynamical boundary conditions considered in [4]. Under certain assumptions of the nonlinear terms we show a regularity result for the unique solution of the problem. We establish a general result about boundedness of invariant sets for the associated evolution process in the norm of the domain of the spatial linear operator appearing in the equation. As a consequence, we deduce that the pullback attractors of the model are bounded in this domain norm. After that, under additional assumptions, some exponential growth results for pullback attractors when time goes to −∞ are proved. Keywords: Reaction-diffusion equations; dynamical boundary conditions; pullback attractors; tempered exponential growth. Mathematics Subject Classifications (2010): 35K57, 35K61, 35B41, 37B55. 1 Introduction and setting of the problem PDE problems with dynamic boundary conditions have the main characteristic of involving the time derivative of the unknown on the boundary of the domain. Although not so often considered in the literature as other boundary conditions (Dirichlet, Neumann, or Robin), dynamic boundary conditions also appear in a wide variety of applied problems. Its use dates back at least to 1901 in the context of heat transfer in a solid in contact with a moving fluid (see the pioneering work by Peddie [36] in 1901, by March and Weaver [34] in 1928, by Peek [37] in 1929, by Langer [28] in 1932, and by Bauer [7] in 1953). From the second half of the 20th century until today, they have been studied in many disciplines. Continuing with the topic of heat transfer between a domain and its boundary, see the more recent references [45, 33] and other variants, like heat transfer in two phase medium ∗Prof. Jos´e Real deceased on 2012. M. A. and P.M.-R. would like to dedicate this paper to his memory. 1 (e.g., cf. [39, 43, 11]; see [22] for a detailed study of a Cahn-Hilliard problem, a natural higher order generalization of the reaction-diffusion equation); problems in fluid dynamics (cf. [21] among others); diffusion phenomena (see [14]), in particular, diffusion in porous medium (e.g., cf. [20], beside the already cited reference [37]); probability theory and mathematical modeling in Biology (cf. [19]); thermoelasticity (cf. [26]); thermal energy storage devices (cf. [1]); chemical engineering (cf. [29]); semiconductor devices (cf. [42]), etcetera (see [8] for more problems and classical references). A detailed physical interpretation of this boundary condition can be found in [25]. For a more abstract point of view the reader is referred to [16, 5, 6] among others. The study of evolution equations with dynamic boundary conditions from the mathematical point of view dates back to 1961, when J. L. Lions [31] treated such equations and gave weak solutions by means of the variational method. Since then, this issue has been investigated to a large extent. In particular, the reaction-diffusion equation with dynamical boundary conditions arises in hydrodynamics and the heat transfer theory (see the above references), and this problem has strong background in mathematical physics. While some of the initial studies were devoted to existence results (e.g., cf. [27, 16, 17, 5, 38]), after that, the borderline (critical exponents) where blow-up phenomena appears or well-posedness holds was analyzed (e.g., cf. [6]), and some others papers are focused directly on well-posedness and the longtime behaviour under suitable dissipativity assumptions, including equilibria, description of convergence, different type of attractors, attracting w.r.t. several metrics, etcetera (cf. [18, 46, 23, 11, 47, 48, 4]). By last in this brief introduction, let us recall that regularity issues concerning problems with dynamical boundary conditions have also been intensively analyzed. This is due to several facts. For instance, regularity properties allow to manage in different ways proofs of existence with different techniques, not only variational, but also by fixed points, using density arguments combined with approximating problems. Regularity features are also useful to deal with different flows in several phase-spaces, to improve attraction properties, and to implement more sophisticated numeric schemes. For example, regularity properties of several problems with dynamical boundary conditions, even of higher order, as wave equations and Cahn-Hilliard equations, with dynamic boundary conditions, have been addressed in [40, 41, 30]. Let us introduce the model we will be involved with in this paper. Let Ω ⊂RNbe a bounded domain with smooth enough boundary. We consider the non-autonomous reaction-diffusion equation ∂u ∂t −∆u+κu +f(u) = h(t) in Ω ×(τ, ∞),(1) with the dynamical boundary condition ∂u ∂t +∂u ∂~n +g(u) = ρ(t) on ∂Ω×(τ, ∞),(2) where ~n is the outer normal to ∂Ω, and the initial conditions u(x, τ) = uτ(x) for x∈Ω,(3) u(x, τ) = ψτ(x) for x∈∂Ω,(4) where τ∈Ris an initial time, and κ > 0, uτ∈L2(Ω), ψτ∈L2(∂Ω), h ∈L2 loc(R;L2(Ω)),and ρ∈L2 loc(R;L2(∂Ω)) are given. 2 We also assume that the functions fand g∈C(R) are given, and satisfy that there exist constants p≥2, q≥2, α1>0, α2>0, β > 0, and l > 0, such that α1|s|p−β≤f(s)s≤α2|s|p+β∀s∈R, α1|s|q−β≤g(s)s≤α2|s|q+β∀s∈R, (f(s)−f(r))(s−r)≥ −l(s−r)2and (g(s)−g(r))(s−r)≥ −l(s−r)2∀s, r ∈R. It is easy to deduce from above that there exists a constant C > 0 such that |f(s)| ≤ C(1 + |s|p−1),|g(s)| ≤ C(1 + |s|q−1) for all s∈R. Let us denote F(s) := Zs 0 f(r)dr and G(s) := Zs 0 g(r)dr. Then, there exist positive constants eα1,eα2,and e βsuch that eα1|s|p−e β≤ F(s)≤eα2|s|p+e β∀s∈R,(5) and eα1|s|q−e β≤ G(s)≤eα2|s|q+e β∀s∈R.(6) As we mentioned before, this model arises in different areas, especially in population growth, chemical reactions and heat conduction. For instance, in the case of a heat transfer in a medium Ω, the equation (1) is a heat equation including an internal heat source. On the other hand, the heat flow from inside Ω to the boundary is −∂u ∂~n . The accumulation rate of heat on the boundary is ∂u ∂t so that, if we consider a source term on ∂Ω, we have the dynamical boundary condition (2) (see the works by Constantin and Escher [13] and by Goldstein [25] for more details). Since the source terms may represent engineering or physic devices during the procedure (or noise perturbations, as in [12]), it is natural that they are time-depending, to indicate the time where they are acting with different intensities. This makes sense to consider some of the approaches of non-autonomous dynamical systems, as will be recalled below. Before to continue with the setting of the problem, let us introduce some notation that will be useful in the sequel. Let us consider the Hilbert space H:= L2(Ω) ×L2(∂Ω), with the natural inner product ((v, φ),(w, ϕ))H= (v, w)Ω+ (φ, ϕ)∂Ω,which in particular induces the norm |(·,·)|Hgiven by |(v, φ)|2 H=|v|2 Ω+|φ|2 ∂Ω,(v, φ)∈H, where we denote by (·,·)Ω(respectively, (·,·)∂Ω) the inner product in L2(Ω) (respectively, in L2(∂Ω)), and by | · |Ω(respectively, | · |∂Ω) the associated norm. We will also denote (·,·)Ω (respectively, (·,·)∂Ω) the inner product in (L2(Ω))N, and the duality product between Lp0(Ω) and Lp(Ω) (respectively, the duality product between Lq0(∂Ω) and Lq(∂Ω)). If r6= 2, we will denote |·|r,Ω(respectively, |·|r,∂Ω) the norm in Lr(Ω) (respectively in Lr(∂Ω)). By 3 k · kΩwe denote the norm in H1(Ω), which is associated to the inner product ((·,·))Ω:= (∇·,∇·)Ω+ (·,·)Ω. We use the notation γ0for the trace operator u7→ u|∂Ω. The trace operator belongs to L(H1(Ω), H1/2(∂Ω)), and we will use kγ0kto denote the norm of γ0in this space. Finally, we will use k · k∂Ωto denote the norm in H1/2(∂Ω),which is given by kφk∂Ω= inf{kvkΩ:γ0(v) = φ}. We remember that with this norm, H1/2(∂Ω) is a Hilbert space. A weak solution (u, ψ)=(u(·;τ, uτ, ψτ), ψ(·;τ, uτ, ψτ)) to the problem (1)–(4) is a pair of functions (u, ψ), satisfying u∈C([τ, ∞); L2(Ω)), ψ ∈C([τ, ∞); L2(∂Ω)), u ∈L2(τ, T ;H1(Ω))∩ Lp(τ, T ;Lp(Ω)) for all T > τ, ψ ∈L2(τ, T;H1/2(∂Ω)) ∩Lq(τ, T ;Lq(∂Ω)) for all T > τ, and γ0(u(t)) = ψ(t),a.e. t∈(τ, ∞),such that d dt(u(t), v)Ω+d dt(ψ(t), γ0(v))∂Ω+ (∇u(t),∇v)Ω+κ(u(t), v)Ω+ (f(u(t)), v)Ω +(g(γ0(u(t))), γ0(v))∂Ω= (h(t), v)Ω+ (ρ(t), γ0(v))∂Ωin D0(τ, ∞), for all v∈H1(Ω)∩Lp(Ω) such that γ0(v)∈Lq(∂Ω),and fulfills the initial conditions u(τ) = uτ and ψ(τ) = ψτ. In Anguiano et al. [4] we discussed several issues concerning problem (1)-(4). We first proved the existence and uniqueness of solution using the monotonicity method, in the same way as in Chueshov and Schmalfuß [12]. A continuous dependence result with respect to initial data was addressed later. Therefore, the bi-parametric family of maps U(t, τ) : H→H, with τ≤t, given by U(t, τ)(uτ, ψτ)=(u(t), ψ(t)), where (u, ψ) = (u(·;τ, uτ, ψτ), ψ(·;τ, uτ, ψτ)) is the unique weak solution to (1)–(4), defines a process on H, which actually is continuous. Then, we studied the asymptotic behaviour of this process in the framework of pullback attractors (as the deterministic analogy to the random attractor considered in [12]). Recall that a pullback attractor for the process U(e.g., cf. [9, 10, 15]) is a family A={A(t) : t∈ R}of compact subsets of Hsuch that it satisfies the non-autonomous “invariance property” U(t, τ)A(τ) = A(t) for all τ≤t; and it attracts in pullback sense elements from a certain given universe, i.e., lim τ→−∞ sup (uτ,ψτ)∈B(τ) inf (v,w)∈A(t)|U(t, τ)(uτ, ψτ)−(v, w)|H= 0 for all t∈R,and for any element {B(τ)}τ∈R⊂ P(H) of the universe. Namely, in [4, Theorem 23] we established the existence of two pullback attractors ADH F and ADH µ(respectively for the universes of fixed bounded sets, and for another, parameterized in time, given by a tempered condition), assuming that hand ρsatisfy Z0 −∞ eµs(|h(s)|2 Ω+|ρ(s)|2 ∂Ω)ds < ∞,(7) where µ∈(0,2λ1), λ1was given by (41) in [4], the first eigenvalue of a linear operator A1 (recalled below). Also, using [35] and under additional assumptions on hand ρwe proved that ADH F(t) = ADH µ(t) for all t∈R.Moreover, these two pullback attractors satisfy the same tempered growth in norm L2(Ω) ×L2(∂Ω) of the second universe, lim τ→−∞ eµτ sup (v,φ)∈A(τ) (|v|2 Ω+|φ|2 ∂Ω)=0.(8) 4 However, as far as we know, there are no results in the literature concerning regularity and exponential growth of the pullback attractors as we will consider in this paper (for similar results for the reaction-diffusion equations with zero Dirichlet boundary condition see [2, 3], and for the Navier-Stokes equations see [24]). As commented before, the regularity results on the solutions and the attractors (that we obtain here) might be useful in the future in order to implement new methods to seek solutions of more general problems by different arguments, to gain attraction in higher norms, or for numerical purposes. The structure of the paper is as follows. In Section 2, under an additional assumption relating the nonlinear terms fand g, we establish a regularity result for the unique solution to problem (1)-(4). This relation among fand g(cf. Theorem 4) forces to take q=p, which will be assumed here on. In Section 3 we prove some results which, in particular, imply that under suitable assumptions, any pullback attractor Asatisfies that any section A(t) or union of sections for a finite time-interval are bounded subsets of (H1(Ω)×H1/2(∂Ω))∩(Lp(Ω)×Lp(∂Ω)) and also in the norm of the domain of A1,D(A1). This result implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data. Finally, in Section 4 we prove two exponential growth results for these pullback attractors, firstly in the norm (H1(Ω) ×H1/2(∂Ω)) ∩(Lp(Ω) ×Lp(∂Ω)), and later, under appropriate additional assumptions, in the norm of D(A1). 2 A regularity result In this section we prove a regularity result for the weak solution to (1)-(4), whose existence and uniqueness is guaranteed in [4]. Let us consider the space V1:= {(v, γ0(v)) : v∈H1(Ω)}. We note that V1is a closed vector subspace of H1(Ω) ×H1/2(∂Ω),and therefore, with the norm k(·,·)kV1given by k(v, γ0(v))k2 V1=kvk2 Ω+kγ0(v)k2 ∂Ω,(v, γ0(v)) ∈V1, V1is a Hilbert space. On the other hand, V1is densely embedded in H. Now, on V1we define a continuous symmetric linear operator A1:V1→V0 1, given by hA1((v, γ0(v))),(w, γ0(w))i= (∇v, ∇w)Ω+κ(v, w)Ω,∀v, w ∈H1(Ω). Remark 1 The domain of the operator A1can be described as follows (which gives sense to the natural definition D(A1) = {(u, γ0(u)) ∈V1:A1((u, γ0(u))) ∈H}). Let (w, ψ)∈H. Assume that (u, γ0(u)) ∈V1solves the equation A1((u, γ0(u))) = (w, ψ). Then (see [5]) (u, γ0(u)), where u∈H1(Ω),is a weak (variational) solution to the problem (−∆u+κu =win Ω, ∂u ∂~n =ψon ∂Ω. 5 Then, we may denote D(A1) = {(v, γ0(v)) ∈V1: ∆ v∈L2(Ω),∂v ∂~n ∈L2(∂Ω)}, with the scalar product (v, w)D(A1)= (A1v, A1w)Hfor any vand w∈D(A1), which induces the norm k(·,·)kD(A1)given by k(v, γ0(v))k2 D(A1)=|A1((v, γ0(v)))|2 H. Let us observe that the space H1(Ω)×H1/2(∂Ω) is compactly imbedded in H, and therefore, for the symmetric and coercive linear continuous operator A1:V1→V0 1, there exists a nondecreasing sequence 0 < λ1≤λ2≤. . . of eigenvalues associated to the operator A1with limj→∞ λj=∞,and there exists a Hilbert basis of H,{(wj, γ0(wj)) : j≥1}⊂ D(A1), with span{(wj, γ0(wj)) : j≥1}densely embedded in V1, such that A1((wj, γ0(wj))) = λj(wj, γ0(wj)) ∀j≥1. Remark 2 It can be proved that span{(wj, γ0(wj)) : j≥1}is densely embedded in V1∩ (Lp(Ω) ×Lp(∂Ω)). Taking into account the above facts, we denote by (um(t), γ0(um(t))) = (um(t;τ, uτ, ψτ), γ0(um(t;τ, uτ, ψτ))) the Galerkin approximation of the solution (u(t;τ, uτ, ψτ), γ0(u(t;τ, uτ, ψτ))) to (1)-(4) for each integer m≥1, which is given by (um(t), γ0(um(t))) = m X j=1 δmj(t)(wj, γ0(wj)),(9) and is the solution of d dt((um(t), γ0(um(t))),(wj, γ0(wj)))H+hA1((um(t), γ0(um(t)))),(wj, γ0(wj))i +(f(um(t)), wj)Ω+ (g(γ0(um(t))), γ0(wj))∂Ω = (h(t), wj)Ω+ (ρ(t), γ0(wj))∂Ω, j = 1, . . . , m, (10) with initial data (um(τ), γ0(um(τ))) = (uτm, γ0(uτm)),(11) where δmj(t)=(um(t), wj)Ω+ (γ0(um(t)), γ0(wj))∂Ω and (uτm, γ0(uτm)) ∈span{(wj, γ0(wj)) : j= 1, . . . , m}converge (when m→ ∞) to (uτ, ψτ) in a suitable sense which will be specified below. Next result gives some preliminar (and standard) uniform estimates for the Galerkin approximations defined above when the initial data are suitably chosen. We include its proof for clarity since it will be used in the sequel. Lemma 3 Under the given assumptions on κ, f,g,h, and ρ, consider a sequence {(uτm, ψτm)} bounded in Hwith each (uτm, ψτm)∈span{(wj, γ0(wj)) : j= 1, . . . , m}.Then, the sequence of corresponding solutions to the Galerkin scheme (9)-(11) with these initial data is bounded in L2(τ, T ;V1)∩Lp(τ, T;Lp(Ω) ×Lp(∂Ω)) for all T > τ. 6 Proof. Multiplying by δmj in (10), and summing from j= 1 to m, we obtain 1 2 d dr|(um(r), γ0(um(r)))|2 H+hA1((um(r), γ0(um(r)))),(um(r), γ0(um(r)))i +(f(um(r)), um(r))Ω+ (g(γ0(um(r))), γ0(um(r)))∂Ω = (h(r), um(r))Ω+ (ρ(r), γ0(um(r)))∂Ω, a.e. r > τ. Using the properties of fand g, −(f(um(r)), um(r))Ω≤ZΩ (β−α1|um(r)|p)dx =β|Ω| − α1|um(r)|p p,Ω∀r≥τ, and −(g(γ0(um(r))), γ0(um(r)))∂Ω≤Z∂Ω (β−α1|γ0(um(r))|p)dσ(x) =β|∂Ω| − α1|γ0(um(r))|p p,∂Ω∀r≥τ. On the other hand, (h(r), um(r))Ω≤1 κ|h(r)|2 Ω+κ 4|um(r)|2 Ω, a.e. r > τ, and as |γ0(um(r))|∂Ω≤ kγ0kkum(r)kΩ, we have (ρ(r), γ0(um(r)))∂Ω≤kγ0k2 2 min{1, κ/2}|ρ(r)|2 ∂Ω+min{1, κ/2} 2kγ0k2|γ0(um(r))|2 ∂Ω ≤kγ0k2 2 min{1, κ/2}|ρ(r)|2 ∂Ω+1 2min{1, κ/2}kum(r)k2 Ω, a.e. r > τ. From above and the inequality (16) of [4], we deduce d dr|(um(r), γ0(um(r)))|2 H+min{1, κ} 1 + kγ0k2k(um(r), γ0(um(r)))k2 V1 +2α1|um(r)|p p,Ω+ 2α1|γ0(um(r))|p p,∂Ω ≤2β(|Ω|+|∂Ω|) + 2 κ|h(r)|2 Ω+kγ0k2 min{1, κ/2}|ρ(r)|2 ∂Ω, a.e. r > τ. (12) Integrating between τand T, the result follows. Now we may establish a regularity result for the solution to the problem. Theorem 4 Suppose that in addition to the assumptions on the data of the problem, we have that p=q≥2,that f,g∈ C1(R)and that there exists a constant cfg >0such that |f(s)−g(s)| ≤ cfg(1 + |s|)∀s∈R,(13) (which is only possible if p=q). Then, for any initial condition (uτ, ψτ)∈V1∩(Lp(Ω) × Lp(∂Ω)), the weak solution to (1)-(4), whose existence and uniqueness is guaranteed in [4], satisfies in addition that (u, ψ)∈C([τ, T]; V1)∩L2(τ, T;D(A1)) ∀T > τ. 7 Proof. We first note that under the conditions imposed we have that f0(s)≥ −l, g0(s)≥ −l∀s∈R.(14) Let (uτ, ψτ)∈V1∩(Lp(Ω)×Lp(∂Ω)). For all m≥1, by Remark 2, there exists (uτm, γ0(uτm)) ∈ span{(wj, γ0(wj)) : 1 ≤j≤m}, such that {(uτm, γ0(uτm))}converges to (uτ, ψτ) in V1and in Lp(Ω) ×Lp(∂Ω) (and therefore in L2(Ω) ×L2(∂Ω)). For each integer m≥1, we consider the sequence {(um, γ0(um))}defined by (9)-(11) with these initial data. In order to prove the result, we will show firstly that the Galerkin approximations are uniformly bounded in L∞(τ, T ;V1∩Lp(Ω) ×Lp(∂Ω)) with derivatives uniformly bounded in L2(τ, T ;H) (and therefore the solution too) for all T > τ. Secondly, we will check that this sequence (and again the solution) is also bounded in L2(τ, T ;D(A1)) for all T > τ, whence the result follows. Step 1. The sequence {(um, γ0(um))}is bounded in L∞(τ, T;V1∩Lp(Ω) ×Lp(∂Ω)) and their derivatives {(u0 m, γ0(u0 m))}are bounded in L2(τ, T ;H) for all T > τ. Multiplying by the derivative δ0 mj in (10), and summing from j= 1 to m, we obtain |(u0 m(r), γ0(u0 m(r)))|2 H+1 2 d dr(hA1((um(r), γ0(um(r)))),(um(r), γ0(um(r)))i) +(f(um(r)), u0 m(r))Ω+ (g(γ0(um(r))), γ0(u0 m(r)))∂Ω = (h(r), u0 m(r))Ω+ (ρ(r), γ0(u0 m(r)))∂Ω, a.e. r > τ. We observe that (f(um(r)), u0 m(r))Ω=d dr ZΩ F(um(r))dx, and (g(γ0(um(r))), γ0(u0 m(r)))∂Ω=d dr Z∂Ω G(γ0(um(r)))dσ(x) for all r≥τ. Then, we deduce |(u0 m(r), γ0(u0 m(r)))|2 H+1 2 d dr(hA1((um(r), γ0(um(r)))),(um(r), γ0(um(r)))i) ≤1 2|h(r)|2 Ω+1 2|u0 m(r)|2 Ω+1 2|ρ(r)|2 ∂Ω+1 2|γ0(u0 m(r))|2 ∂Ω −d dr ZΩ F(um(r))dx −d dr Z∂Ω G(γ0(um(r)))dσ(x), a.e. r > τ. (15) Integrating now between τand t, and taking into account the definition of A1(cf. inequality (16) of [4]), (5) and (6), we obtain that Zt τ |(u0 m(θ), γ0(u0 m(θ)))|2 Hdθ +min{1, κ} 1 + kγ0k2k(um(t), γ0(um(t)))k2 V1 +2eα1|um(t)|p p,Ω+ 2eα1|γ0(um(t))|p p,∂Ω ≤max{1, κ}k(um(τ), γ0(um(τ)))k2 V1+Zt τ (|h(θ)|2 Ω+|ρ(θ)|2 ∂Ω)dθ +2eα2|um(τ)|p p,Ω+ 2eα2|γ0(um(τ))|p p,∂Ω+ 4˜ β(|Ω|+|∂Ω|) 8 for all τ≤t. As long as the sequence {(um(τ), γ0(um(τ)))}is bounded in V1and in Lp(Ω) ×Lp(∂Ω), Step 1 is proved. Moreover, taking into account the uniqueness of solution to (1)-(4) and using Ascoli-Arzel`a Theorem, it is not difficult to conclude that the sequence {(um, γ0(um))}converges weakly-star in L∞(τ, T ;Lp(Ω) ×Lp(∂Ω)) and in L∞(τ, T;V1) to the solution (u, γ0(u)) to (1)-(4), and we also obtain that (u0, γ0(u0)) ∈L2(τ, T ;H). Step 2. The sequence {(um, γ0(um))}is bounded in L2(τ, T ;D(A1)). In this way, by the above arguments we will have that (u, γ0(u)) ∈L2(τ, T;D(A1)). After that, as (u0, γ0(u0)) ∈L2(τ, T;H), by the energy equality (see [44]), we can deduce that (u, γ0(u)) ∈C([τ, T ]; V1) and the theorem will be proved. Multiplying in (10) by λjδmj , where λjis the eigenvalue associated to the eigenfunction (wj, γ0(wj)), and summing once more from j= 1 to m, we obtain (A1((um(r), γ0(um(r)))),(u0 m(r), γ0(u0 m(r))))H+|A1((um(r), γ0(um(r))))|2 H +(f(um(r)), m X j=1 λjδmj(r)wj)Ω+ (g(γ0(um(r))), m X j=1 λjδmj(r)γ0(wj))∂Ω = (h(r), m X j=1 λjδmj(r)wj)Ω+ (ρ(r), m X j=1 λjδmj(r)γ0(wj))∂Ω, a.e. r > τ. (16) It follows from the properties of f, (14), and integrating by parts, that −(f(um(r)), m X j=1 λjδmjwj)Ω =−ZΩ f(um(x, r))(−∆um(x, r) + κum(x, r))dx =−ZΩ f0(um(x, r))|∇um(x, r)|2dx+Z∂Ω f(γ0(um(x, r)))∂um ∂~n dσ(x)−κ ZΩ f(um(x, r))um(x, r)dx ≤l|∇um(r)|2 Ω+Z∂Ω f(γ0(um(x, r)))∂um ∂~n dσ(x)−κα1|um(r)|p p,Ω+κβ|Ω| ∀r≥τ. On the other hand, we have −(g(γ0(um(r))), m X j=1 λjδmj(r)γ0(wj))∂Ω=−Z∂Ω g(γ0(um(x, r)))∂um ∂~n dσ(x)∀r≥τ, whence, jointly with the last inequality, −(f(um(r)), m X j=1 λjδmj(r)wj)Ω−(g(γ0(um(r))), m X j=1 λjδmj(r)γ0(wj))∂Ω ≤l|∇um(r)|2 Ω−κα1|um(r)|p p,Ω+κβ|Ω|+Z∂Ω (f(γ0(um(x, r))) −g(γ0(um(x, r))))∂um ∂~n dσ(x) for all r≥τ. 9 for all m≥1. From these two inequalities, we obtain |(u0 m(r), γ0(u0 m(r)))|2 H≤(4l+ 3) max{1, κ}k(um(τ+ 1), γ0(um(τ+ 1)))k2 V1 +(4l+ 3) Zτ+3 τ (|h(θ)|2 Ω+|ρ(θ)|2 ∂Ω)dθ + 4e β(|Ω|+|∂Ω|) +2eα2(4l+ 3)(|um(τ+ 1)|p p,Ω+|γ0(um(τ+ 1))|p p,∂Ω) +Zτ+3 τ+1 (|h0(θ)|2 Ω+|ρ0(θ)|2 ∂Ω)dθ for all r∈[τ+ 2, τ + 3], and any m≥1. Using this inequality and (25), it yields |(u0 m(r), γ0(u0 m(r)))|2 H ≤C4|(uτ, ψτ)|2 H+Zτ+3 τ (|h(θ)|2 Ω+|h0(θ)|2 Ω+|ρ(θ)|2 ∂Ω+|ρ0(θ)|2 ∂Ω)dθ + 1(29) for all r∈[τ+ 2, τ + 3], and any m≥1, where C4= (4l+ 3)(C3max{1, κ, 2eα2}+ max{1,4e β(|Ω|+|∂Ω|)}). From inequality (17), we obtain |A1((um(r), γ0(um(r))))|2 H≤4|(u0 m(r), γ0(u0 m(r)))|2 H+ 4 max{l, 4c2 fgc2 ∂Ω}kum(r), γ0(um(r))k2 V1 +4κβ|Ω|+ 16c2 fg|∂Ω|+ 8(|h(r)|2 Ω+|ρ(r)|2 ∂Ω), where we have denoted c∂Ωthe injection constant of H1/2(∂Ω) in L2(∂Ω). Using this inequality, and thanks to (25) and (29), we obtain |A1((um(r), γ0(um(r))))|2 H ≤4C4|(uτ, ψτ)|2 H+Zτ+3 τ (|h(θ)|2 Ω+|h0(θ)|2 Ω+|ρ(θ)|2 ∂Ω+|ρ0(θ)|2 ∂Ω)dθ + 1 +4 max{l, 4c2 fgc2 ∂Ω}C3|(uτ, ψτ)|2 H+Zτ+3 τ (|h(θ)|2 Ω+|ρ(θ)|2 ∂Ω)dθ + 1 +4κβ|Ω|+ 16c2 fg|∂Ω|+ 8(|h(r)|2 Ω+|ρ(r)|2 ∂Ω) for all r∈[τ+ 2, τ + 3], and any m≥1. On the other hand, by Theorem 5 we have that (u(·;τ, uτ, ψτ), γ0(u(·;τ, uτ, ψτ))) ∈C([τ+ 2, τ + 3]; V1). Since we know that {(um(·;τ, uτ, ψτ), γ0(um(·;τ, uτ, ψτ)))}converges weakly to (u(·;τ, uτ, ψτ), γ0(u(·;τ, uτ, ψτ))) in L2(τ+ 2, τ + 3; D(A1)), then, using Lemma 6, and the above inequality, we have that |A1((u(r), γ0(u(r))))|2 H≤C5|(uτ, ψτ)|2 H+Zτ+3 τ (|h(θ)|2 Ω+|h0(θ)|2 Ω+|ρ(θ)|2 ∂Ω+|ρ0(θ)|2 ∂Ω)dθ + sup r∈[τ+2,τ+3] (|h(r)|2 Ω+|ρ(r)|2 ∂Ω)+1(30) 16 for all r∈[τ+ 2, τ + 3], any τ∈R, and (uτ, ψτ)∈H, where C5= 4 max{2, C4+C3max{l, 4c2 fgc2 ∂Ω}+κβ|Ω|+ 4c2 fg|∂Ω|}. Now, observe that by the Cauchy-Schwartz inequality, |h(r)|Ω≤ |h(τ+ 2)|Ω+Zτ+3 τ+2 |h0(θ)|2 Ωdθ1/2 and |ρ(r)|∂Ω≤ |ρ(τ+ 2)|∂Ω+Zτ+3 τ+2 |ρ0(θ)|2 ∂Ωdθ1/2 for all r∈[τ+ 2, τ + 3]. Thus, from (30) and the above inequalities, we deduce that kU(τ+ 2, τ)(uτ, ψτ)k2 D(A1)≤3C5|(uτ, ψτ)|2 H+|h(τ+ 2)|2 Ω+|ρ(τ+ 2)|2 ∂Ω +Zτ+3 τ (|h(θ)|2 Ω+|h0(θ)|2 Ω+|ρ(θ)|2 ∂Ω+|ρ0(θ)|2 ∂Ω)dθ + 1 for all τ∈R, and (uτ, ψτ)∈H. From this inequality and the fact that A(τ) = U(τ, τ −2)A(τ−2), we obtain k(v, w)k2 D(A1)≤3C5sup (ϕ,ψ)∈A(τ−2) |(ϕ, ψ)|2 H+|h(τ)|2 Ω+|ρ(τ)|2 ∂Ω +Zτ+1 τ−2 (|h(θ)|2 Ω+|h0(θ)|2 Ω+|ρ(θ)|2 ∂Ω+|ρ0(θ)|2 ∂Ω)dθ + 1 for all (v, w)∈ A(τ), and any τ∈R. Now, from this last inequality, and thanks to (7), (8), (26) and (27), we obtain (28). Remark 15 Theorems 13 and 14 apply to the tempered pullback attractor ADH µand therefore also (trivially) to ADH Fsince ADH F(t)⊂ ADH µ(t)for all t∈R. Acknowledgments This work is dedicated to the memory of Prof. Jos´e Real, with our deepest and most sincere admiration, gratitude, and love. He was M.A. and P.M.-R.’s PhD-advisor, a clever mathematician with a sharp view on problems, a generous and wonderful colleague, and an even better friend. He passed away too soon, being only 60 years old. We miss him deeply, but he will stay forever in our hearts. The authors thank the referee for observations that led us to improve the presentation of the manuscript. 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