A Non-Autonomous Strongly Damped Wave Equation: Existence and Continuity of the Pullback Attractor

A NON-AUTONOMOUS STRONGLY DAMPED WAVE EQUATION: EXISTENCE AND CONTINUITY OF THE PULLBACK ATTRACTOR TOM´ AS CARABALLO1, ALEXANDRE N. CARVALHO2, JOS´ E A. LANGA1, AND FELIPE RIVERO1 Abstract. In this paper we consider the strongly damped wave equation with time dependent terms utt −∆u−γ(t)∆ut+βε(t)ut=f(u), in a bounded domain Ω ⊂Rn, under some restrictions on βε(t), γ(t) and growth restrictions on the non-linear term f. The function βε(t) depends on a parameter ε,βε(t)ε→0 −→ 0. We will prove, under suitable assumptions, local and global well posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {Aε(t) : t∈R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ǫ= 0. 1. Introduction Let us consider the equation (1) utt −∆u−γ(t)∆ut+βε(t)ut=f(u) in Ω, u= 0 on ∂Ω, in a sufficiently smooth bounded domain Ω ⊂Rn,γ, βε:R→(0,∞) verifies 0 < γ0⩽γ(t)⩽ γ1<∞, 0 < β0ε⩽β(t)⩽β1ε<∞, and γ(t) and βε(t) are continuously differentiable in R and with bounded derivative uniformly in ǫ. We also suppose that βiε ε→0 −→ 0,for i= 0,1 β1ε β0ε ε→0 −→ 1. (2) For the nonlinearity f:R→Rwe assume that it verifies the following dissipativeness and growth conditions f∈C2(R,R),lim sup |s|→∞ f(s) s⩽0,(3) |f′′(s)|⩽c(1 + |u|ρ−2)(4) where 1 < ρ < n+2 n−2. 1Partially supported by Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088 and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. 2Partially supported by CNPq 305447/2005-0 and 451761/2008-1, CAPES/DGU 267/2008 and FAPESP 2008/53094-4, Brazil; Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088, and Junta de Andaluc´ıa grant # P07-FQM-02468, 1 2 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO The aim of this paper is to obtain the existence, regularity, continuity, characterization and continuity of characterization of pullback attractors for the problem (1) in H1 0(Ω) ×L2(Ω). In particular, for the characterization of the pullback attractors we obtain that (when all the equilibria are hyperbolic) all solutions in the pullback attractors are backwards and forwards asymptotic to equilibria. We also obtain that the evolution process associated to (1) in H1 0(Ω) ×L2(Ω) is a gradient-like evolution process (homoclinic structures do not exist). There are some relevant physical applications for this kind of equation. In space dimensions n= 1 and n= 2, if γ(t)≡1 and βε(t)≡0, equation (1) models the longitudinal vibrations of a homogeneous bar subjected to viscous effects. The term −∆utindicates that the stress is proportional to the strain rate as in a linearized Kelvin-Voigt material. In dimension three, the model describes the variation from the configuration at rest of a homogeneous and isotropic linear viscoelastic solid with short memory, called rate type (see [17]), in the presence of an external displacement-dependent force. The term βutwith β > 0 indicates that the bar is subjected to dynamical friction as well. We also have a perturbed sine-Gordon equation of the form utt −α∆ut−∆u+ sin u+βut=f(u), describing the evolution of the current uin a Josephson junction (see [16]). This equation is a generalization of an autonomous model which has been investigated in many articles by several authors (see, for example, [6, 7, 3, 13, 14, 21, 24, 25]), with particular emphasis on its asymptotic behaviour. Assuming (critical) growth restrictions on the non-linear term f, the global existence of solutions in the autonomous case has been established in [6], based on the theory of ε-regular solutions. To better explain the results in the paper we introduce the terminology of evolution processes. Our framework will be that of evolution processes for non-autonomous dynamical systems (Cheban [12], Chepyzhov-Vishik [15], Kloeden [22], Sell [26]). An evolution process in a Banach space Zis a family of maps {S(t, s) : t⩾s}from Zinto itself with the following properties: 1) S(t, t) = I, for all t∈R, 2) S(t, s) = S(t, τ)S(τ, s),for all t⩾τ⩾s, 3) {(t, s)∈R2:t⩾s} × Z ∋ (t, s, x)7→ S(t, s)x∈ Z is continuous. A family of sets {K(t) : t∈R}pullback attracts bounded subsets of Zunder {S(t, s) : t⩾ s}if K(t) pullback attracts all bounded subsets at tunder {S(t, s) : t⩾s}, for each t∈R, i.e., for each bounded subset C⊂ Z, lim s→−∞ dist(S(t, s)C, K(t)) = 0, where dist(·,·) denotes the Hausdorff semi-distance, dist(A, B) = sup a∈A inf b∈B|a−b|. An evolution process {S(t, s) : t⩾s}is said to be pullback strongly bounded dissipative if, for each t∈R,there exists a bounded subset B(t) of Zwhich pullback attracts bounded subsets of Zat time τ, for each τ⩽t. In other words, given a bounded subset Bof Zand τ⩽t, there exists s0(τ, D) such that S(τ, s)B⊂B(t), for all s⩽s0(τ, D). A NON-AUTONOMOUS STRONGLY DAMPED WAVE EQUATION 3 A family {A(t) : t∈R}of compact subsets of Zis said to be the pullback attractor for the evolution process {S(t, s) : t⩾s}if it is invariant (S(t, s)A(s) = A(t) for all s⩽t), pullback attracts bounded subsets of Z, and is minimal, that is, if there is another family of closed sets {C(t) : t∈R}which pullback attracts bounded subsets of Z, then A(t)⊆C(t), for all t∈R. In Section 2 we will use the theory of uniform sectorial operators, as in [11, 27], to prove the local well-posedness and regularity for (1) in H1 0(Ω) ×L2(Ω). The existence of the attractor is proved in Section 3 using results based on asymptotic compact processes that appear in [4] and using an energy functional to obtain some useful estimates. In Section 4, following the ideas in [5] (see also [25]), we show that the pullback attractor is a bounded set in H2(Ω) ×H2(Ω). In Section 5 it is shown how the pullback attractor of the limit problem is a gradient-like attractor, that is, all the bounded global solutions converge backwards and forwards to some fixed point of the system, and then the attractor can be characterized as the union of unstable manifolds of equilibria. Finally, in Section 6 we will prove the continuity of the attractors when ε→0. To this end, we first show the upper-semicontinuity using standard techniques based on the continuity of the processes, and then follow [9] (see also [8, 10]) to obtain the lower semicontinuity. One can find some particular cases containing time-dependent damped terms, for example, in [20, 23] (see also the references therein). 2. Existence of solution To obtain the local well posedness of the Cauchy problem for (1) we will use the results in [11] (see also, [27]) which we introduce next Definition 2.1. Let Zbe a Banach space and B(t) : D⊂ Z → Z (Dfixed) a closed, densely defined, time-dependent operator. a) The operator family B(t)is called uniformly sectorial if there is a constant C > 0 (independent of t∈R) such that k(λI +B(t))−1kL(Z)⩽C |λ|+ 1;∀λ∈Cwith Re λ⩾0. b) B(t)is called uniformly H¨older continuous if there exist constants C > 0and ε > 0 such that, for any t, τ ,s∈R, k[B(t)−B(τ)] B−1(s)kL(Z)⩽C(t−τ)ε. Consider the Cauchy problem zt=B(t)z+F(z), t > t0, z(t0) = z0. (5) Definition 2.2. For a continuous function F:R× Zα→ Z,α∈[0,1),z(·, t0, z0) : [t0, t0+τ)→ Zαis a solution for (5) if it is continuous, continuously differentiable in (0, τ), z(t, t0, z0)∈D(B(t)) for t∈(t0, t0+τ)and (5) is satisfied for all t∈(0, τ). We can now state the following result (see [11] for a more general version that includes the critical growth case). 4 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO Theorem 2.3. If the operator family B(t)is uniformly sectorial and uniformly H¨older continuos, and F:Zα→ Z is Lispschitz continuous in bounded subsets of Zαthen, given r > 0there exists τ > 0and for each z0∈ Zαwith kz0kZα⩽ra function z(·, t0, z0)∈ C([t0, t0+τ],Zα)∩C1((t0, t0+τ],Zα)with the properties that {z0∈ Zα:kz0kZα⩽r} ∋ z07→ z(·, t0, z0)∈C([t0, t0+τ],Zα) is continuous, z(·, t0, z0)is the unique solution of (5). It is clear from this result (since the time of existence can be chosen uniform in bounded subsets of Zα) that solutions that do not blow up in the Zαmust exist for all t⩾t0. Next we establish the functional analytic framework needed to apply the above results to the problem (1). Let Ω ⊂Rnbe a bounded smooth domain, X=L2(Ω) and A: D(A)⊂X→Xbe the positive self-adjoint operator defined by D(A) = H2(Ω) ∩H1 0(Ω) and Au =−∆ufor all u∈D(A). Let Aα, with α∈R, be the fractional power operators associated to Aand, for α⩾0, let Xα=D(Aα), with the norm k · kα=kAα· kX, be the fractional power spaces associated to Afor α⩾0. Define X−α,α > 0, as the completion of Xwith respect to the norm kA−α· kX. With this notation, we will keep denoting by Athe closed extension of Ato X−α,α⩾0. Finally, denote by Y(−1) =X1 2×X−1 2 With this, we can view (1) as the system (6) u vt +A(t)u v=Fu v where D(A(t)) = X1 2×X1 2, A(t)φ ϕ=0−I A γ(t)A+βε(t)φ ϕ:= −ϕ A(φ+γ(t)ϕ) + βǫ(t)ϕ, and Fu z=0 f(u), where, for φ: Ω →R,f(φ) is defined by f(φ)(x) = f(φ(x)), x∈Ω. Note that D(A(t)) (as a vector space) does not depend on t. Denote by Y1 (−1) the space X1 2×X1 2with the norm kA(t)· kY(−1) , where (7) A(t) = 0−I A γ(t)A+βǫ(t)I. It is easy to see that Y1 (−1) is isomorphic to X1 2×X1 2with its usual norm, uniformly for t∈R. Also note that, A(t)−1=γ(t)I+βǫ(t)A−1A−1 −I0. Denote by Yα (−1) the domain of A(t0)αfor some t0∈Rand α⩾0 with the graph norm; we obtain the fractional power scale spaces associated to Aα −1(t0) (see[18]). As in [6], we can prove that the following embeddings hold (the operator A(t) is acretive and, as a consequence A NON-AUTONOMOUS STRONGLY DAMPED WAVE EQUATION 5 of that, it will have bounded imaginary powers, and hence the fractional power spaces will coincide with the interpolation spaces): (8)    Yα (−1) ⊂H1(Ω) ×H2α−1(Ω) ⊂Lq1(Ω) ×Lq2(Ω), for 1 ⩽q1⩽2n n−2,1⩽q2⩽2n n−2(2α−1), α ∈[0,1] , n ⩾3. Using this we establish the following local well posedness result for (6). Theorem 2.4. The operator family A(t)is uniformly sectorial and uniformly H¨older continuos, and F:R×Y 1 2 (−1) →Y(−1) defined by Fu v=0 f(u)is Lispschitz continuous in bounded subsets of Y 1 2 (−1). Then, given r > 0there exists τ > 0and for each U0∈Y 1 2 (−1) with kU0kY 1 2 (−1) ⩽ra function U(·, t0, U0)∈C([t0, t0+τ], Y 1 2 (−1))∩C1((t0, t0+τ], Y 1 2 (−1))with the properties that {U0∈Y 1 2 (−1) :kU0kY 1 2 (−1) ⩽r} ∋ U07→ U(·, t0, U0)∈C([t0, t0+τ], Y 1 2 (−1)) is continuous, U(·, t0, U0)is the unique solution of (6) (in the sense of Definition 2.2) satisfying U(t0, t0, U0) = U0. If U0∈X1×X1, then U(·)is twice continuously differentiable in (0, τ)with values in X1 2×X. The proof of this theorem consists on the verification of the assumptions of Theorem 2.3 and the proof of the the time regularity in the last statement of it. First we prove the uniform sectoriality and the uniform H¨older continuity of A(t). First note that (λ+A(t))−1=((λ+βǫ(t))I+γ(t)A)R(λ)R(λ) −AR(λ)λR(λ) where R(λ) = (λ(λ+βǫ(t))I+ (1 + λγ(t))A)−1. With this, it is easy to check that  (λ+A(t))−1 L(Y(−1))⩽C |λ|+ 1. Now, [A(t)−A(s)]A(t0)−1=−(γ(t)−γ(s)) 0 0 A0−(βǫ(t)−βǫ(s)) 0 0 I0. From this and from the H¨older continuity of γand βǫuniformly in Rand in ǫ, it is clear that k[A(t)−A(s)]A(t0)−1kL(Y(−1))⩽M|t−s|ǫ which proves the uniform H¨older continuity condition b). Lemma 2.5. If f:R→Rsatisfies (4) then, kf(w1)−f(w2)kL2n n+2 ⩽ckw1−w2kH11 + kw1kρ−1 H1+kw2kρ−1 H1, w1, w2∈H1(Ω). (9) 6 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO Proof: From (4), the H¨older inequality, and the Sobolev embeddings we obtain: kf(w1)−f(w2)kL2n n+2 (Ω) ⩽cZΩ|w1−w2|(1 + |w1|ρ−1+|w2|ρ−1)2n n+2 n+2 2n ⩽ckw1−w2kL2n n−2(Ω)k1 + |w1|ρ−1+|w2|ρ−1kLn 2(Ω) ⩽˜ckw1−w2kL2n n−2(Ω) 1 + kw1kρ−1 Ln(ρ−1) 2(Ω) +kw2kρ−1 Ln(ρ−1) 2(Ω). Since 2n(ρ−1) 4⩽2n n−2, we have that Ln(ρ−1) 2(Ω) ⊃H1(Ω). Which completes the proof. This proves the following result. Lemma 2.6. If f:R→Rsatisfies the above conditions then, kFφ1 ψ1−Fφ2 ψ2kY(−1) ⩽ckφ1 ψ1−φ2 ψ2kY 1 2 (−1) ⩽c 1 + kφ1 ψ1kρ−1 Y 1 2 (−1) +kφ2 ψ2kρ−1 Y(−1) !,φ1 ψ1,φ2 ψ2∈Y 1 2 (−1). (10) From all this and from Theorem 2.3 follows all assertions of Theorem 2.4 except the last one. To see that if U0∈X1×X1,U(·, U0)∈C2((0, τ), X 1 2×X) we apply again Theorem 2.3 to the following Cauchy problem d dt   u v w +A(t)  u v w =¯ F  u v w , t > t0   u(t0) v(t0) w(t0) =  u0 v0 −Au0−γ′(t0)Av0−βǫ(t0)v0+f(u0)  with Z=X1×X1×X,A(t) : D(A(t)) ⊂ Z → Z,D(A(t)) = X1×X1×X1 A(t) =   0−I0 0 0 −I 0 (1 + γ′(t))A+β′ ǫ(t)I γ(t)A+βǫ(t)I and ¯ F  u v w =  0 0 ¯ f(u, v)  A NON-AUTONOMOUS STRONGLY DAMPED WAVE EQUATION 7 with f(u, v)(x) := f′(u(x))v(x). Firstly, we need to rewrite the problem since A(t) is singular and is not invertible. Therefore, we consider d dt   u v w +C(t)  u v w =¯ H  u v w , with C(t) =   −I0 0 0 0 −I 0 (N+γ′(t))A+β′ ǫ(t)I γ(t)A+βǫ(t)I and ¯ H  u v w =  u−v 0 ¯ h(u, v)  with h(u, v)(x) := (1 + N)Av +f′(u(x))v(x) and Nbig enough. Then, there is a unique solutions which is a continuously differentiable function with values in Z=X1×X1×X since ¯ h:Z → Z is Lypschitz in bounded subsets of Zand C(t) easily satisfies all properties of Theorem 2.3. Also, using techniques similar to those used to study fin Lemma 2.6, we obtain properties of the nonlinearity ¯ h:X1×X1×X→Xthat ensure that ¯ Halso satisfies the hypothesis of Theorem 2.3 and proves the last statement of Theorem 2.4 Remark 2.7. Note that, if u∈X1 2and following Lemma 2.5, f(u)∈L2n n+2−2ǫ֒→X−1 2+ǫfor some ǫ > 0such that (11) 0 < ǫ =n(2 −ρ) 2+ρ. Hence, we may obtain that the solutions given in Theorem 2.4 with initial data in X1 2+ǫ×Xǫ are in fact in X1 2+ǫ×X1 2+ǫfor t∈(t0, t0+τ). We can repeat this reasoning, noting that if u∈X1 2+ǫ, then f(u)∈X−1 2+ρǫ and, after a finite number of steps, we obtain that solutions with initial data in X1×X1 2are in X1×X1. 3. Existence of pullback attractors First we need to recall some definitions and properties which will be needed in this section. An evolution process {S(t, s) : t⩾s}in a Banach space Zis said pullback asymptotically compact if, for each t∈R, each sequence {sk}k∈Nin (−∞, t] such that sk k→∞ −→ −∞, and each bounded sequence {zk}k∈Nin Zwith {S(t, sk)zk:k∈N}bounded, the sequence {S(t, sk)zk}k∈Npossesses a convergent subsequence. It is said that {S(t, s) : t⩾s}is pullback strongly bounded if, for each t∈Rand each bounded subset Bof Z,Ss⩽tSτ⩽sS(s, τ)Bis bounded. Now we state the followings results from [5], which will lead to the existence of the pullback attractor for (6). Theorem 3.1. If an evolution process {S(t, s) : t⩾s}is pullback strongly bounded dissipative and pullback asymptotically compact, then {S(t, s) : t⩾s}possesses a pullback attractor {A(t) : t∈R}with the property that Ss⩽tA(s)is bounded for each t∈R. Theorem 3.2. Let {S(t, s) : t⩾s}be a pullback strongly bounded process in a Banach space Zsuch that S(t, s) = T(t, s) + U(t, s), where U(t, s)is compact and there exists a non-increasing function k:R+×R+−→ Rwith k(σ, r)→0when σ→ ∞, and for all s⩽t 8 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO and z∈Zwith kzk⩽r,kT(t, s)zk⩽k(t−s, r). Then the process {S(t, s) : t⩾s}is pullback asymptotically compact. From now on, we denote by Y0:= H1 0(Ω) ×L2(Ω), which will be the phase space or our process (notice that with the notation used in Section 2, Y0=X1/2×Xwhich is isomorphic to Y 1 2 (−1)). For b∈R+, let us define Lb(·,·) : Y0−→ Ras (12) Lb(φ, ϕ) = 1 2kφk2 H1 0+1 2kϕk2 L2+b(φ, ϕ)L2−ZΩ G(φ)dx, with G(s) = Rs 0f(θ)dθ. If b 2λ−1 1⩽1 4and b⩽1 2, we have that 1 4[kϕk2 H1 0+kφk2 L2]⩽1 2kϕk2 H1 0+b(ϕ, φ)L2+1 2kφk2 L2⩽3 4[kϕk2 H1 0+kφk2 L2]. Following [19], it follows from (3), for each δ > 0 there is a constant Cδ>0 such that ZΩ f(u)u dx ⩽δkuk2 L2(Ω) +Cδ, ZΩ G(u)dx ⩽δkuk2 L2(Ω) +Cδ for each u∈L2(Ω) such that f(u)u∈L1(Ω) and G(u)∈L1(Ω). Also proceeding as in the proof of Proposition 4.1 in [1], there is a constant c0>0 and given r > 0 there is a constant crsuch that (13) ZΩ G(u)dx ≤crkuk2 H1 0 ZΩ f(u)u dx ≤crkuk2 H1 0        for all u∈H1 0(Ω) with kukH1 0(Ω) ⩽r. From this, if band δare suitably small, there is a constant K > 0 such that Lb(φ, ϕ)⩾1 8hkϕk2 H1 0+kφk2 L2i−K and for each r > 0 a constant Kr>0 such that Lb(φ, ϕ)⩽Krhkϕk2 H1 0+kφk2 L2i for all (φ, ϕ)∈Y0such that k(φ, ϕ)kY0⩽r. Thanks to the regularity of the solutions (see Theorem 2.4), if (u, ut) denotes the solution of (6) with (u(t0), ut(t0) = (u0, v0), we can differentiate the expression Lb(u, ut), A NON-AUTONOMOUS STRONGLY DAMPED WAVE EQUATION 9 d dtLb(u, ut) = (utt −∆u−f(u), ut)L2+bkutk2 L2 +b(u, ∆u+γ(t)∆ut−βε(t)ut+f(u))L2 ⩽−γ0kutk2 H1 0−β0εkutk2 L2−bγ(t)(u, ut)H1 0−bkuk2 H1 0 −bβε(t)(u, ut)L2+b(u, f(u))L2+bkutk2 L2 ⩽−γ0 2kutk2 H1 0−λ1γ0 2kutk2 L2−β0εkutk2 L2−bkuk2 H1 0 +bγ11 2a1 kuk2 H1 0+a1 2kutk2 H1 0+bβ1ε1 2a2 kuk2 L2+a2 2kutk2 L2 +bkutk2 L2+b(δkuk2 H1 0+Cδ). (14) Hence, for suitable choice of a1,a2,band δ, there is a C0>0 such that (15) d dtLb(u, ut)⩽−C0kuk2 H1 0+kutk2 L2+bCδ. From (14) with b= 0, given r > 0 (16) Br:=sup nkuk2 H1 0(Ω) +kutk2 L2(Ω) :t⩾t0, u ∈H1 0(Ω) with kukH1 0(Ω),kutkL2(Ω) ⩽ro<∞. Now, if k(u(t), ut(t))k2 Y0>bCδ C0+1 C0=r2 0for all t⩾t0, then there exists a time Tr>0 such that Lb(u(t), ut(t)) ⩽0 for each t⩾t0+Tr. In other case, there exists tusuch that k(u(tu), ut(tu))k2 Y0⩽bCδ C0+1 C0=r2 0(and let tube the smallest time with this property), and then k(u(t), ut(t))kY0⩽R1for all t⩾t0+tu, where R1is the radius of a ball containing the set Br0(defined in (16)). If tu⩽Tr, then k(u(t), ut(t))kY0⩽R1for all t⩾t0+Tr. If tu> Tr, then Lb(u(t), ut(t)) ⩽0 for t∈[t0+Tr, t0+tu] and k(u(t), ut(t))kY0⩽R1for all t⩾t0+tu. This implies that the ball of radius R0pullback strongly absorbs bounded subsets of Y0 where R2 0= max{R2 1,8K}. Let us now consider the process {S(t, s); t⩾s}generated by our problem in the phase space Y0. Recall that S(t, s) is given as follows. For each initial value w0:= (u0, v0)∈Y0and each initial time s∈R, system (6) possesses a unique solution which can be written as (17) S(t, s)w0=T(t, s)w0+U(t, s)w0=u(t, s, w0) ut(t, s, w0) , where T(t, s) is the evolution process associated to the linear part of (6)(i.e. for f= 0), and (18) U(t, s)w0=Zt s T(t, τ)F(S(τ, s)w0)dτ. Proceeding as in (14) and using the functional ˜ Lb(u, ut) = 1 2kuk2 H1 0+1 2kutkL2+b(u, ut)L2 being now (u, ut) a solution of the linear part of (6), we can prove that this linear part decays exponentially to zero. The compactness of Ufollows from the following facts. 16 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO We then need to prove that all the equilibrium points in Eare hyperbolic for all Sε(t, s). If u(t) is a solution of (30) and defining z(t) = u(t)−e∗ jfor any e∗ j∈ E, then z(·) satisfies (45) (ztt −∆z−γ(t)∆zt−f′(e∗ j)z=h(z) z(s) = z0;zt(s) = z1, where h(z) = f(z+e∗ j)−f(e∗ j)−f′(e∗ j)z. Then, we can construct the following system (46) z xt +¯ A0(t)z x=Hz x where ¯ A0(t) = 0−I −∆−f′(e∗ j)−γ(t)∆ and Hz x=0 h(z). In the same way, taking v(t) solution of (1) and defining z(t) = v(t)−e∗ jwe have (47) z xt +¯ A0(t) + Bε(t)z x=Hz x where Bε(t) = 0 0 0βε(t), with H(0) ≡0 and the Jacobian matrix JH(0) ≡0∈ L(Y0). Let {Z0(t, s) : t⩾s}and {Zε(t, s) : t⩾s}denote the processes associated to (46) and (47) respectively. Note that sup t∈R kBε(t)kL(Y0)= sup t∈R |βε(t)|⩽β1ε ε→0 −→ 0, then, by Theorem 7.6.11 in [18], it follows that for ε0>0 sufficiently small, {Zε(t, s) : t⩾s} has an exponential dichotomy and, consequently, every point in Eis also hyperbolic for {Sε(t, s) : t⩾s}, for all ε < ε0. Therefore we can apply Theorem 6.2, obtaining the lower-semicontinuity of the pullback attractors. Remark 6.3. Observe that, in particular, we have proved upper and lower semicontinuity of pullback attractors with respect to the autonomous strongly damped wave equation, i.e., the one with γ(t) = γ > 0. Acknowledgements. We would like to thank the referee for his/her interesting remarks and suggestions. References [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Communications in Partial Differential Equations,17 (5-6) (1992) 841-866. [2] A.B. Babin, M.I. 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Series 2, Vol. 49: Ten Papers on Functional Analysis and Measure Theory, American Mathematical Society, Providence, R.I., (1966), 162. (T. Caraballo) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain 18 T. CARABALLO, A. N. CARVALHO, J. A. LANGA, AND F. RIVERO (A. N. Carvalho) Instituto de Ciˆ encias Matem´ aticas e de Computac¸ao, Universidade de S˜ ao Paulo-Campus de S˜ ao Carlos, Caixa Postal 668, 13560-970 S˜ ao Carlos SP, Brazil (J. A. Langa) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain (F. Rivero) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain