Pullback exponential attractors for parabolic equations with dynamical boundary conditions

TAIWANESE JOURNAL OF MATHEMATICS Vol. 21, No. 4, pp. 819–839, August 2017 DOI: 10.11650/tjm/7862 Pullback Exponential Attractors for Parabolic Equations with Dynamical Boundary Conditions Rados law Czaja* and Pedro Mar´ın-Rubio Abstract. The existence of pullback exponential attractors for a nonautonomous semilinear parabolic equation with dynamical boundary condition is proved when the timedependent forcing terms are translation bounded or even grow exponentially in the past and in the future. 1. Introduction In this paper we consider the nonautonomous semilinear parabolic equation with dynamical boundary condition of the form (1.1)                  ∂u ∂t −∆u+κu +f1(u) = h1(t) in Ω ×(s, ∞), ∂u ∂t +∂u ∂~n +f2(u) = h2(t) on ∂Ω×(s, ∞), u(x, s) = us(x) for x∈Ω, u(x, s) = ϕs(x) for x∈∂Ω, where Ω is a bounded domain in RN,N≥2, with a Lipschitz boundary ∂Ω, ~n is the outer normal unit vector to ∂Ω, s∈Ris an initial time, us,ϕsare initial data, κ > 0, and the functions f1,f2,h1,h2are given. Parabolic equations of the above type with dynamical boundary conditions serve as models in the heat transfer theory and in hydrodynamics, for example in the description of the heat transfer in a solid body in contact with a moving fluid. They have been investigated in many research articles (e.g., see [1–3, 11] and the references therein). Received October 24, 2016; Accepted November 9, 2016. Communicated by Yingfei Yi. 2010 Mathematics Subject Classification. Primary: 37B55; Secondary: 35B41, 35K58. Key words and phrases. pullback attractors, exponential attractors, semilinear parabolic equations, dynamical boundary conditions. The second author was partially supported by Ministerio de Educaci´on–DGPU through project PHB2010- 0002-PC, and projects MTM2015-63723-P and P12-FQM-1492, Spain. *Corresponding author. 819 820 Rados law Czaja and Pedro Mar´ın-Rubio We assume that us∈L2(Ω), ϕs∈L2(∂Ω), h1∈L2 loc(R;L2(Ω)), h2∈L2 loc(R;L2(∂Ω)), and the functions f1, f2∈C(R) satisfy the following assumptions (fi(u)−fi(v))(u−v)≥ −l(u−v)2, u, v ∈R, i = 1,2,(1.2) |fi(u)−fi(v)| ≤ L|u−v|1 + |u|pi−2+|v|pi−2, u, v ∈R, i = 1,2,(1.3) fi(u)u≥α|u|pi−β, u ∈R, i = 1,2,(1.4) with some constants pi≥2, α, l, L > 0, β≥0. The above conditions on the nonlinearities make that equations in problem (1.1) become a reaction-diffusion equation with dynamical boundary conditions. Note that, in particular, as fiwe may take fi(u) = u|u|pi−2−u,u∈R, with pi>2. We also see that (1.2) means that the functions R3u7→ fi(u) + lu ∈R,i= 1,2, are nondecreasing. Moreover, we observe that there exists C > 0 such that (1.5) |fi(u)| ≤ C(1 + |u|pi−1), u ∈R, i = 1,2. Finally, if p1=p2= 2, then (1.3) implies global Lipschitz continuity of fi,i= 1,2, i.e., (1.6) |fi(u)−fi(v)| ≤ e L|u−v|, u, v ∈R, i = 1,2, and the condition in (1.2) holds with l=e L= 3L. Remark 1.1.If the system (1.1) does not contain the term with κ, but (1.4) holds, then by a suitable change of f1, it can be considered in the form of (1.1) with any positive κ for p1>2 and 0 <κ<αfor p1= 2. Indeed, define e f1(u) = f1(u)−κu and note that (1.2) implies e f1(u)−e f1(v)(u−v)≥ −(l+κ)(u−v)2, u, v ∈R, and, if p1>2, for every ε > 0 there exists cε>0 such that e f1(u)u≥(α−ε)|u|p1−β−cε, u ∈R, whereas if p1= 2 we have e f1(u)u≥(α−κ)|u|2−β, u ∈R. Moreover, (1.3) implies e f1(u)−e f1(v)≤(L+κ)|u−v|1 + |u|p1−2+|v|p1−2, u, v ∈R. In [1], under assumptions (1.2) and (1.4) for ~ f= (f1, f2) and under some extra integrability condition for ~ h= (h1, h2), the authors proved the existence of an evolution Pullback Exponential Attractors for PE with Dynamical BC 821 process for (1.1) on the space H=L2(Ω) ×L2(∂Ω), which possesses a minimal pullback attractor. A minimal pullback attractor {A(t) : t∈R}for a process {U(t, s) : t≥s}on a Banach space Eis a family of nonempty compact subsets of E, which is invariant under the process, i.e., U(t, s)A(s) = A(t) for t≥s, it pullback attracts all bounded subsets of E, i.e., for any bounded subset Dof Eand t∈R lim s→∞ distE(U(t, t −s)D, A(t)) = 0, where distE(A, B) = supx∈Ainfy∈Bkx−ykEdenotes the Hausdorff semidistance in E, and satisfies a minimality condition, which guarantees its uniqueness: if another family {C(t) : t∈R}of nonempty closed subsets of Epullback attracts all bounded subsets of E, then A(t)⊂C(t) for t∈R. In the present article our aim is to prove the existence of a pullback exponential attractor for (1.1). This family {M(t) : t∈R}of nonempty compact subsets of Eis only positively invariant under the process, i.e., U(t, s)M(s)⊂ M(t) for t≥s, but we require that the fractal dimension in E(denoted by dimE f(·)) of the sets forming the family has a uniform bound, i.e., there exists d≥0 such that sup t∈R dimE f(M(t)) ≤d < ∞, and the pullback attraction of bounded subsets of Etowards M(t) is at an exponential rate. This means that there exists ω > 0 such that for every bounded subset Dof Eand t∈Rwe have lim s→∞ eωs distE(U(t, t −s)D, M(t)) = 0. Note that the existence of a pullback exponential attractor {M(t) : t∈R}implies the existence of the minimal pullback attractor {A(t) : t∈R}as its subset, that is, A(t)⊂ M(t) for t∈R. In particular, the minimal pullback attractor also has a uniform bound of the fractal dimension. The first constructions of pullback exponential attractors were presented in [8–10,14,16] and later in [5]. In this paper, however, we use the recent results of [7] to show the existence of pullback exponential attractors. In Section 4 we prove the existence of a pullback exponential attractor for (1.1) in H=L2(Ω) ×L2(∂Ω) (cf. Theorem 4.5) if the forcing term ~ h= (h1, h2)∈L2 loc(R;H) is translation bounded, i.e., there exists K > 0 such that (1.7) sup t∈RZt+1 t~ h(τ) 2 Hdτ ≤K, and the nonlinear terms fi,i= 1,2, have suitable exponents pi(see (4.5)) due to the available a priori estimate in H. If an additional condition (4.11) is satisfied, we are able 822 Rados law Czaja and Pedro Mar´ın-Rubio to consider higher exponents p1=p2=pgiven in (4.15). In particular, for N= 2 the nonlinearities fi(u) = u3−u,u∈R, among many others, are admitted. In Section 5 we consider the Lipschitz case (p1=p2= 2) and show in Theorem 5.4 the existence of a pullback exponential attractor for (1.1) in Heven if the time-dependent forcing terms h1and h2may grow exponentially in the past and in the future, i.e., when the function ~ h= (h1, h2)∈L2 loc(R;H) admits the exponential growth (1.8) ~ h(t) 2 H≤Keθ|t|, t ∈R for some K > 0 and 0 ≤θ < 2(λ1+α), where λ1>0 is the first eigenvalue of the operator A0, specified in (2.5). 2. Evolution process of global weak solutions We consider the problem (1.1) with us∈L2(Ω), ϕs∈L2(∂Ω), h1∈L2 loc(R;L2(Ω)) and h2∈L2 loc(R;L2(∂Ω)) given. Moreover, we assume that fi∈C(R), i= 1,2, satisfy (1.2)–(1.4). We denote by |·|p,Ω(respectively, |·|p,∂Ω) the norm in Lp(Ω) (respectively, in Lp(∂Ω)) and by (·,·)Ω(respectively, (·,·)∂Ω) the inner product in L2(Ω) and (L2(Ω))N, which defines the norm |·|2,Ω=|·|Ω, and the duality product between Lp0(Ω) and Lp(Ω) (respectively, the inner product in L2(∂Ω), which defines the norm |·|2,∂Ω=|·|∂Ω, and the duality product between Lp0(∂Ω) and Lp(∂Ω)). The notation |·| will also be used for the Lebesgue measure of a set in both RNor RN−1, without more indications since no confusion arises. By k·kΩwe denote the norm in H1(Ω), which is associated to the inner product ((·,·))Ω= (∇·,∇·)Ω+ (·,·)Ω. Furthermore, γ0will denote the trace operator γ0(u) = u|∂Ω, u ∈C∞(Ω), which belongs to L(H1(Ω), H1/2(∂Ω)) with norm kγ0kand is surjective. The norm in the subspace H1/2(∂Ω) of L2(∂Ω) is given by kuk1/2,∂Ω= Z∂Ω|u(x)|2dσx+ZZ∂Ω×∂Ω |u(x)−u(y)|2 |x−y|Ndσxdσy!1/2 and makes H1/2(∂Ω) a Hilbert space. Moreover, H1 0(Ω) = u∈H1(Ω) : γ0(u)=0, H1/2(∂Ω) is a dense subspace of L2(∂Ω) and γ0maps bounded subsets of H1(Ω) into relatively compact subsets of L2(∂Ω) (for details see [12, Chapter 1], [13, Chapter 6] and [17, Chapter 2]). Finally, let us observe that throughout the paper BE r(x) denotes the Pullback Exponential Attractors for PE with Dynamical BC 823 open ball in a metric space Eof center xand radius r, and clEAdenotes the closure in the topology of Eof a certain subset Aof E. Following [1, 15] we will show existence and uniqueness of global weak solutions of (1.1). Definition 2.1. A global weak solution of (1.1) is a pair of functions (u, ϕ) satisfying u∈C([s, ∞); L2(Ω)) ∩L2(s, T;H1(Ω)) ∩Lp1(s, T;Lp1(Ω)), ϕ∈C([s, ∞); L2(∂Ω)) ∩L2(s, T;H1/2(∂Ω)) ∩Lp2(s, T;Lp2(∂Ω)) for all T > s,γ0(u(t)) = ϕ(t) for a.e. t∈(s, ∞), the following equality holds for all v∈H1(Ω) ∩Lp1(Ω) such that γ0(v)∈Lp2(∂Ω) d dt(u(t), v)Ω+d dt(ϕ(t), γ0(v))∂Ω+ (∇u(t),∇v)Ω+κ(u(t), v)Ω + (f1(u(t)), v)Ω+ (f2(γ0(u(t))), γ0(v))∂Ω = (h1(t), v)Ω+ (h2(t), γ0(v))∂Ωfor a.e. t∈(s, ∞), and u(s) = usand ϕ(s) = ϕs. As in the proof of [1, Theorem 5] we introduce the following spaces (with corresponding norms) and the following operators, which will be useful in the sequel. We define a Hilbert space H=L2(Ω) ×L2(∂Ω), with the inner product ((u, ϕ),(v, ψ))H= (u, v)Ω+ (ϕ, ψ)∂Ω, which induces the norm |·|Hgiven by |(u, ϕ)|2 H=|u|2 Ω+|ϕ|2 ∂Ωfor (u, ϕ)∈H, and the closed vector subspace of H1(Ω) ×H1/2(∂Ω) defined as V0=(u, γ0(u)) : u∈H1(Ω) with the norm given by k(u, γ0(u))k2 V0=kuk2 Ω+kγ0(u)k2 1/2,∂Ωfor (u, γ0(u)) ∈V0. Observe that V0is a Hilbert space, which is densely and compactly embedded in H. We identify Hwith its dual by the Riesz theorem and therefore we have the chain of inclusions V0⊂ H⊂V0 0. We consider the continuous linear operator A0:V0→V0 0defined through a symmetric continuous bilinear form B:V0×V0→Rgiven as B[~u,~v] = hA0~u,~viV0 0,V0= (∇u, ∇v)Ω+κ(u, v)Ω, where ~u = (u, γ0(u)),~v = (v, γ0(v)) ∈V0, since (2.1) |B[~u,~v]| ≤ (1 + κ)k~ukV0k~vkV0, ~u,~v ∈V0. 824 Rados law Czaja and Pedro Mar´ın-Rubio Recall that Bis coercive (cf. (16) in [1]), i.e., (2.2) B[~u, ~u]≥1 1 + kγ0k2min {1, κ}k~uk2 V0, ~u ∈V0. By Lax-Milgram lemma there exists the bounded inverse A−1 0:V0 0→V0. Its restriction to His a bounded compact operator, which is the inverse of the unbounded linear operator A0:H⊃D(A0)→Hwith the domain D(A0) = {~u ∈V0:A0~u ∈H}. This operator is symmetric and surjective. Moreover, it is positive, since for ~u = (u, γ0(u)) ∈D(A0) we have (A0~u, ~u)H=hA0~u, ~uiV0 0,V0=|∇u|2 Ω+κ|u|2 Ω≥min {1, κ/2}min n1,kγ0k−2o|~u|2 H. Hence there exists an orthonormal basis {~wj= (wj, γ0(wj))} ⊂ D(A0) in the Hilbert space Hconsisting of eigenfunctions of A0, with corresponding eigenvalues λjsuch that λj+1 ≥λj>0, j∈N, and λj→ ∞. We define the linear subspaces E0={~ 0}and (2.3) En= span {~w1, . . . , ~wn}, n ∈N, of V0and note that the bilinear form Bdefines an inner product in V0and ~wj/ pλjis an orthonormal basis in V0with this inner product. Consequently, for any ~u ∈V0such that ~u ⊥En−1,we have B[~u, ~u] = ∞ X j=1 λ−1 jB[~u, ~wj]2= ∞ X j=n λj(~u, ~wj)2 H≥λn|~u|2 H, n ∈N. Hence we obtain (2.4) λn= min ~u∈V0\{0} ~u⊥En−1 hA0~u, ~uiV0 0,V0 |~u|2 H , n ∈N. In particular, we have (2.5) λ1= min ~u∈V0\{0} hA0~u, ~uiV0 0,V0 |~u|2 H . Now, we introduce the nonlinear operators A1:V1→V0 1and A2:V2→V0 2given by A1(u, ϕ) = (f1(u),0),(u, ϕ)∈V1=Lp1(Ω) ×L2(∂Ω), A2(u, ϕ) = (0, f2(ϕ)),(u, ϕ)∈V2=L2(Ω) ×Lp2(∂Ω). The operators are well-defined by (1.5). Note that Vi,i= 0,1,2, are separable, reflexive Banach spaces, densely embedded in H. We define V= 2 \ i=0 Vi=V0∩(Lp1(Ω) ×Lp2(∂Ω)) with k~uk2 V= 2 X i=0 k~uk2 Vi. Pullback Exponential Attractors for PE with Dynamical BC 825 We see that Vis a separable Banach space, densely embedded in H. Thus, we have V⊂H⊂V0and Vi⊂H⊂V0 i, i = 0,1,2. Observe that from (1.3) it follows that each Ai,i= 0,1,2, is hemicontinuous, i.e., for every ~u,~v, ~w ∈Vithe function R3µ7→ hAi(~u +µ~v), ~wiV0 i,Vi∈R is continuous. Moreover, by (1.5) we see that kAi(~u)kV0 i≤Ci1 + k~ukpi−1 Vi, ~u = (u, ϕ)∈Vi, i = 1,2. We also have by (2.1) kA0~ukV0 0≤(1 + κ)k~ukV0, ~u ∈V0. By (2.2) and (1.2) each operator is monotone, i.e., hA0(~u −~v), ~u −~viV0 0,V0≥0, ~u,~v ∈V0, hAi(~u)−Ai(~v), ~u −~viV0 i,Vi≥ −l|~u −~v|2 H, ~u,~v ∈Vi, i = 1,2. Finally, we have by (1.4) hA1(~u), ~uiV0 1,V1≥α|u|p1 p1,Ω−β|Ω|, ~u = (u, ϕ)∈V1, hA2(~u), ~uiV0 2,V2≥α|ϕ|p2 p2,∂Ω−β|∂Ω|, ~u = (u, ϕ)∈V2, and by (2.2) hA0(~u), ~uiV0 0,V0≥1 1 + kγ0k2min {1, κ}k~uk2 V0, ~u ∈V0. Then by a modification of [15, Chapter 2, Theorem 1.4] for every ~ h= (h1, h2)∈L2 loc(R;H), s∈R,T > s and ~us= (us, ϕs)∈Hthere exists a unique function ~u ∈L2(s, T ;V0)∩Lp1(s, T ;V1)∩Lp2(s, T ;V2)∩C([0, T ], H) such that        d~u dt + 2 X i=0 Ai(~u) = ~ h, ~u(s) = ~us. Moreover, we obtain the energy equality for a.e. t>s 1 2 d dt |~u(t)|2 H+ 2 X i=0 hAi(~u(t)), ~u(t)iV0 i,Vi= (~ h(t), ~u(t))H. Thus we have proved (cf. also [1, Theorem 5]) the result on the existence and uniqueness of the global weak solutions to (1.1). 826 Rados law Czaja and Pedro Mar´ın-Rubio Theorem 2.2. Under conditions (1.2)–(1.4) for any s∈R,(us, ϕs)∈L2(Ω) ×L2(∂Ω) there exists a unique global weak solution (u, ϕ)of problem (1.1). Moreover, this solution satisfies the energy equality 1 2 d dt |u(t)|2 Ω+|ϕ(t)|2 ∂Ω+|∇u(t)|2 Ω+κ|u(t)|2 Ω+ (f1(u(t)), u(t))Ω + (f2(ϕ(t)), ϕ(t))∂Ω = (h1(t), u(t))Ω+ (h2(t), ϕ(t))∂Ω (2.6) for a.e. t>s. Some conclusions from the above functional setting, abstract formulation and energy equality are given below. The first one is that the global weak solutions of (1.1) satisfy the following differential inequality. Proposition 2.3. Under the assumptions of Theorem 2.2, the solution ~u = (u, ϕ)of (1.1) satisfies with any δ > 0 (2.7) d dt |~u(t)|2 H+ (2λ1−δ)|~u(t)|2 H≤2β(|Ω|+|∂Ω|) + δ−1~ h(t) 2 H for a.e. t>s. Proof. We apply (1.4) and (2.5) to (2.6) to get d dt |~u(t)|2 H+ 2λ1|~u(t)|2 H≤2β(|Ω|+|∂Ω|) + 2 [(h1(t), u(t))Ω+ (h2(t), ϕ(t))∂Ω] for a.e. t>s. The Cauchy-Schwarz and Cauchy inequalities lead to (2.7). Another consequence, now from Theorem 2.2, is that the global weak solutions to (1.1) define an evolution process {U(t, s) : t≥s}in H, i.e., (2.8) U(t, s)(us, ϕs) = (u(t), ϕ(t)),(us, ϕs)∈H, where (u, ϕ) is the unique global weak solution of (1.1) with (u(s), ϕ(s)) = (us, ϕs). Observe that the process is Lipschitz continuous on H, which means that for each pair (t, s), the map U(t, s) is Lipschitz (and the Lipschitz constant is not supposed to be uniform for all the pairs). Proposition 2.4. Under the assumptions of Theorem 2.2, for every t≥sthere exists a constant Lt,s =e(l−λ1)(t−s)>0such that |U(t, s)~us−U(t, s)~vs|H≤Lt,s |~us−~vs|H, ~us,~vs∈H. Pullback Exponential Attractors for PE with Dynamical BC 827 Proof. Consider a pair of initial data ~us,~vs∈H. Denoting the corresponding solutions by ~u and ~v, we see that the difference ~w =~u −~v satisfies for a.e. t>s 1 2 d dt |~w|2 H+hA0~w, ~wiV0 0,V0+hA1(~u)−A1(~v), ~wiV0 1,V1+hA2(~u)−A2(~v), ~wiV0 2,V2= 0. Using (1.2) and (2.5), we obtain d dt |~w(t)|2 H+ 2(λ1−l)|~w(t)|2 H≤0 for a.e. t > s. In particular, we conclude |~w(t)|2 H≤e2(l−λ1)(t−s)|~w(s)|2 H, t ≥s, which proves the claim. 3. Existence of exponential pullback attractors Our aim now is to prove the existence of a pullback exponential attractor for the process {U(t, s) : t≥s}in Hdefined in (2.8). To achieve this goal we are going to apply [7, Corollaries 2.6 and 2.8], which we recall below. Theorem 3.1. Let {U(t, s) : t≥s}be a Lipschitz continuous process on a Hilbert space H. Assume that (H1)there exists a family of nonempty closed bounded subsets B(t)of H,t∈R, which is positively invariant under the process, i.e., U(t, s)B(s)⊂B(t), t ≥s, (H2)there exist t0∈R,γ0≥0and M > 0such that diamH(B(t)) < Me−γ0t, t ≤t0, (H3)in the past the family {B(t) : t∈R}pullback absorbs all bounded subsets of H; that is, for every bounded subset Dof Hand t≤t0there exists TD,t ≥0such that U(t, t −r)D⊂B(t), r ≥TD,t, and, additionally, the function (−∞, t0]3t7→ TD,t ∈[0,∞)is nondecreasing for every bounded D⊂H. Next, we assume that the semi-process {U(t, s) : t0≥t≥s}can be represented as U(t, s) = C(t, s) + S(t, s), where {C(t, s) : t0≥t≥s}and {S(t, s) : t0≥t≥s}are families of operators satisfying the following properties: 834 Rados law Czaja and Pedro Mar´ın-Rubio The process possesses also the minimal pullback attractor {A(t) : t∈R}in H, which is contained in the pullback exponential attractor {M(t) = Mν(t) : t∈R}and thus has uniformly bounded fractal dimension. Note that the above result holds for example for the nonlinearities fiof the form fi(u) = u3−aiu,u∈R, for N= 2 under the assumption of the same order of f1and f2, i.e., (4.11) and sufficiently smooth boundary. Actually, many other nonlinearities are allowed, like any polynomial of odd degree with positive leading coefficient. This also shows that the regular minimal pullback attractor obtained in [2] has uniformly bounded fractal dimension in Hif the forcing terms ~ h= (h1, h2) are translation bounded. 5. Exponentially growing forcing terms We consider now (1.1) under assumptions (1.6) and (1.4) with p1=p2= 2. Note that (1.2) holds with l=e L. We will find a pullback absorbing family for the problem (1.1) when the function ~ h= (h1, h2)∈L2 loc(R;H) admits an exponential growth in the past and in the future by assuming (1.8) for some K > 0 and 0 ≤θ < 2(λ1+α), where λ1>0 is the first eigenvalue of the operator A0. Applying (1.4) and (2.5) to the energy equality (2.6), we see that the global weak solutions ~u = (u, ϕ) of (1.1) satisfy for a.e. t>s d dt |~u(t)|2 H+ 2(λ1+α)|~u(t)|2 H≤2β(|Ω|+|∂Ω|) + 2 [(h1(t), u(t))Ω+ (h2(t), ϕ(t))∂Ω]. Hence by the Cauchy inequality for δ > 0 such that 0 < θ +δ < 2(λ1+α) we have for a.e. t>s (5.1) d dt |~u(t)|2 H+ (2(λ1+α)−δ)|~u(t)|2 H≤2β(|Ω|+|∂Ω|) + δ−1~ h(t) 2 H. Using (1.8) and applying the Gronwall inequality to (5.1) we get |~u(t)|2 H≤ |~u(s)|2 He−(2λ1+2α−δ)(t−s)+ 2β(|Ω|+|∂Ω|)(2λ1+ 2α−δ)−1 +δ−1KZt s e−(2λ1+2α−δ)(t−τ)eθ|τ|dτ, t ≥s. Estimating the last term, we obtain (5.2) |~u(t)|2 H≤ |~u(s)|2 He−(2λ1+2α−δ)(t−s)+K1+K2eθ|t|, t ≥s, where K1= 2β(|Ω|+|∂Ω|)(2λ1+ 2α−δ)−1and K2= 2δ−1(2λ1+ 2α−δ−θ)−1K. We define e B(t) = n~u ∈H:|~u|2 H≤2K1+ 2K2eθ|t|o, t ∈R. Pullback Exponential Attractors for PE with Dynamical BC 835 It follows from (5.2) that for every bounded subset Dof Hthere exists rD>0 such that U(t, t −r)D⊂e B(t), r ≥rD, t ∈R. Moreover, there exists r0>0 such that U(t, t −r)e B(t−r)⊂e B(t), r ≥r0, t ∈R, since, by using (5.2), it suffices to check that 2K1e−(2λ1+2α−δ)r+ 2K2eθ|t−r|e−(2λ1+2α−δ)r≤K1+K2eθ|t|, t ∈R, r ≥r0. Thus, the sets (5.3) B(t) = clH[ r≥r0 U(t, t −r)e B(t−r)⊂e B(t), t ∈R form a positively invariant family consisting of nonempty closed bounded subsets of H, which shows (H1). Moreover, we have diamH(B(t)) ≤2p2K1+ 2K2eθ|t|<5 max npK1,pK2oe−θ 2t, t ≤0, so (H2) holds with M= 5 max √K1,√K2,γ0=θ/2 and t0≤0 arbitrary. Furthermore, if Dis a bounded subset of Hand t≤t0, then setting TD=rD+r0and taking s≥TD we get U(t, t −s)D⊂B(t), which shows that (H3) is satisfied in this case. We have proved the following Proposition 5.1. Under assumptions (1.6) and (1.4) with p1=p2= 2 for fi,i= 1,2and ~ h= (h1, h2)∈L2 loc(R;H)satisfying (1.8), the family B(t)defined by (5.3) is positively invariant and pullback absorbing for the process {U(t, s) : t≥s}in H. Moreover, this family satisfies the assumptions (H1)–(H3)in Theorem 3.1. We consider the projections Pn:H→En,Qn=I−Pnas in (4.4). Proposition 5.2. Suppose that fi,i= 1,2, satisfy (1.6) and (1.4) with p1=p2= 2. Assume further that ~ h= (h1, h2)∈L2 loc(R;H)satisfies (1.8). Then the semi-process {U(t, s) : t0≥t≥s}corresponding to problem (1.1) can be decomposed as U(t, s) = QnU(t, s) + PnU(t, s) in such a way that for every 0<ε<2λ1we have |Qn(U(t, s)~us−U(t, s)~vs)|2 H ≤ e−(2λn+1−ε)(t−s)+ε−1e L2 2λn+1 −ε+ 2e Le2 e L(t−s)!|~us−~vs|2 H (5.4) for all t≥sand ~us,~vs∈H. 836 Rados law Czaja and Pedro Mar´ın-Rubio Proof. Let us denote by ~u = (u, ϕ), ~v = (v, ψ) the global weak solutions of (1.1) corresponding to initial data ~us,~vs∈H, respectively. Then, setting ~w =~u −~v and ~z = Qn~w = (I−Pn)~w, we obtain (4.8) as in the proof of Proposition 4.2. Using (2.4) and Cauchy-Schwarz and Cauchy inequalities to (4.8), we get for every 0 < ε < 2λ1and for a.e. t>s d dt |~z|2 H+ (2λn+1 −ε)|~z|2 H≤ε−1|f1(u)−f1(v)|2 Ω+|f2(ϕ)−f2(ψ)|2 ∂Ω. Since fi,i= 1,2, are globally Lipschitz continuous, it follows from (1.6) that d dt |~z|2 H+ (2λn+1 −ε)|~z|2 H≤ε−1e L2|~w|2 Hfor a.e. t > s. By Proposition 2.4, in particular we have d dt e(2λn+1−ε)t|~z(t)|2 H≤ε−1e L2e(2λn+1−ε)t+2 e L(t−s)|~w(s)|2 Hfor a.e. t > s. Integrating and using |~z(s)|H≤ |~w(s)|H, we get (5.4). Corollary 5.3. Under the assumptions of Proposition 5.2, there exist two families of operators {C(t, s) : t0≥t≥s}and {S(t, s) : t0≥t≥s}with U(t, s) = C(t, s) + S(t, s) satisfying hypotheses (H4)–(H5)of Theorem 3.1. Proof. We put et > 0 arbitrary and C=QnUand S=PnUwith some n∈Nlarge enough. (H4) follows from Proposition 5.2 and (H5) follows from [4, Lemma 1] (see also [7, Lemma 4.2]) and Proposition 2.4. In particular, if n∈Nis such that (5.5) λ= e−(2λn+1−ε) e t+ε−1e L2 2λn+1 −ε+ 2e Le2 e L e t!1/2 <1 2e−θ 2 e t, then, for 0 < ν < min n1 2e−θ 2 e t−λ, e( e L−λ1) e to, we have (5.6) Nν≤1 + 2e( e L−λ1) e t νn in (H5). Collecting the above results, as an application of Theorem 3.1, we obtain Theorem 5.4. If fi,i= 1,2, satisfy (1.6) and (1.4) with p1=p2= 2, whereas ~ h= (h1, h2)∈L2 loc(R;H)satisfies (1.8) with some K > 0and 0≤θ < 2(λ1+α), then the process {U(t, s) : t≥s}on H=L2(Ω)×L2(∂Ω) of global weak solutions of (1.1) possesses a pullback exponential attractor {M(t) = Mν(t) : t∈R}in Hsatisfying the properties: (a) M(t)is a nonempty compact subset of B(t)for t∈R, Pullback Exponential Attractors for PE with Dynamical BC 837 (b) U(t, s)M(s)⊂ M(t),t≥s, (c) supt∈RdimH f(M(t)) ≤ −ln Nν/ln (2(ν+λ)) + θ 2et, where λis given in (5.5) and Nνis given in (5.6) for 0< ν < min n1 2e−θ 2 e t−λ, e( e L−λ1) e to, (d) for any t∈Rthere exists ct>0such that for any s≥max {t−t0,0}+ 2et distH(U(t, t −s)B(t−s),M(t)) ≤cte−ω0s, where ω0=−1 e tln (2(ν+λ)) + θ 2et>0, (e) for any 0< ω < ω0we have lim s→∞ eωs distH(U(t, t −s)D, M(t)) = 0, t ∈R,Dbounded in H. The process possesses also the minimal pullback attractor {A(t) : t∈R}in H, which is contained in the pullback exponential attractor {M(t) = Mν(t) : t∈R}and thus has uniformly bounded fractal dimension. It would be interesting to know if we may obtain the existence of pullback exponential attractors or minimal pullback attractors with uniformly bounded fractal dimension when the time-dependent forcing terms grow exponentially, but the nonlinearities have superlinear growth. Acknowledgments The authors thank Jan Cholewa for inspiring conversations and helpful suggestions during the preparation of the paper. References [1] M. Anguiano, P. Mar´ın-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl. 383 (2011), no. 2, 608–618. https://doi.org/10.1016/j.jmaa.2011.05.046 [2] ,Regularity results and exponential growth for pullback attractors of a nonautonomous reaction-diffusion model with dynamical boundary conditions, Nonlinear Anal. Real World Appl. 20 (2014), 112–125. https://doi.org/10.1016/j.nonrwa.2014.05.003 [3] J. M. Arrieta, P. Quittner and A. Rodr´ıguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Differential Integral Equations 14 (2001), no. 12, 1487–1510. 838 Rados law Czaja and Pedro Mar´ın-Rubio [4] F. Balibrea and J. Valero, On dimension of attractors of differential inclusions and reaction-diffusion equations, Discrete Contin. Dynam. Systems 5(1999), no. 3, 515– 528. https://doi.org/10.3934/dcds.1999.5.515 [5] A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal. 12 (2013), no. 6, 3047–3071. https://doi.org/10.3934/cpaa.2013.12.3047 [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications 49, American Mathematical Society, Providence, RI, 2002. [7] R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Anal. 104 (2014), 90–108. https://doi.org/10.1016/j.na.2014.03.020 [8] R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems, J. Math. Anal. Appl. 381 (2011), no. 2, 748–765. https://doi.org/10.1016/j.jmaa.2011.03.053 [9] M. Efendiev, Y. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative system, J. Math. Soc. Japan 63 (2011), no. 2, 647–673. https://doi.org/10.2969/jmsj/06320647 [10] M. Efendiev, S. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A135 (2005), no. 4, 703–730. https://doi.org/10.1017/s030821050000408x [11] Z.-H. Fan and C.-K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal. 68 (2008), no. 6, 1723–1732. https://doi.org/10.1016/j.na.2007.01.005 [12] P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman Publishing, 1985. [13] A. Kufner, O. John, S. Fuˇc´ık, Function Spaces, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. [14] J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst. 26 (2010), no. 4, 1329–1357. https://doi.org/10.3934/dcds.2010.26.1329 [15] J.-L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Gauthier-Villars, Paris, 1969. Pullback Exponential Attractors for PE with Dynamical BC 839 [16] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations 4, 103-200, Handb. Differ. Equ., Elsevier, 2008. https://doi.org/10.1016/s1874-5717(08)00003-0 [17] J. Neˇcas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-10455-8 Rados law Czaja Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland E-mail address:[email protected] Pedro Mar´ın-Rubio Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail address:[email protected]