Pullback and Forward Attractors for a Damped Wave Equation with Delays

PULLBACK AND FORWARD ATTRACTORS FOR A DAMPED WAVE EQUATION WITH DELAYS TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Abstract. The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of Sine-Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model. 1. Introduction We aim to show in this paper how the theory of pullback attractors can be useful in the analysis of the asymptotic behaviour of damped wave equations subject to some kind of hereditary characteristics. For partial differential equations with constant delay, the standard theory of global attractors can be successfully applied to handle this problem (see, Hale [9], Chueshov [6], etc ). However, when the hereditary characteristics are described by terms containing variable or distributed delays, the problem becomes nonautonomous. Although nonautonomous equations can be studied from several points of view (theory of skew product flows [16], kernel sections [5]), we will use the theory of pullback attractors for two-parameter semigroups or processes for several reasons. In particular, it allows us to obtain results for a wide class of delayed terms in a unified way without requiring compactness of the base space as in the skew product formalism or even the formulation of such a base space (see [4]). In [3], it has been developed a program to prove that 2D-Navier-Stokes models with external forces containing bounded delays have pullback and forward attractors. The analysis there does not work in the present situation, as also happens in the non-delay case, since the hyperbolic structure of the equation only enables one to prove the existence of a compact attracting set instead of a compact absorbing set. This is not a consequence of the delay term, but rather of the hyperbolic nature of the equation (see [18]). The delay term considered here, however, is responsible of the nonautonomous character of the problem. To show how the technique works, we will consider a model with a general functional term as an external force which, in particular, includes the models of Sine-Gordon type with delayed forcing term. Such equations, in which the nonlinearity is of the form sin u, occur frequently in applications, and the technical steps needed to handle it are typical of other types of nonlinear terms (although it is complicated to handle a general nonlinearity which include them all). Since our primary aim is to investigate the effect of the delay, we restrict attention to globally Lipschitz nonlinearities like this representative one (see Section 3.4 for more details). Date: June 8, 2004 . Key words and phrases. Wave equations, pullback attractor, forward attractor, variable delay, distributed delay (2000) Mathematics Subject Classification: 35R10, 35B40, 47H20, 58F39, 73K70. Partly supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community), projects HA2001-0075 and BFM2002-03068. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL In Section 2 we set our problem in a suitable nonautonomous framework, and recall some necessary concepts and results from the theory of pullback attractors. The existence of a pullback and a forward attractor is established in Section 3. Finally we include some technical but useful results in an Appendix. 2. Statement of the problem and preliminaries 2.1. The model. Let Ω ⊂Rn, n ≥1,be an open and bounded subset with smooth boundary ∂Ω = Γ, and consider the following problem (1)              ∂2u ∂t2+α∂u ∂t −∆u=f+h(t, ut), t > τ u|Γ= 0, t ≥τ−r, u(x, t) = φ(x, t −τ), x ∈Ω, t ∈[τ−r, τ], ∂u ∂t (x, t) = ∂φ ∂t (x, t −τ), x ∈Ω, t ∈[τ−r, τ], where f+h(t, ut) is the source intensity which may depend on the history of the solution, αis a positive constant, φis the initial datum on the interval [τ−r, τ] where r > 0,and utis defined for θ∈[−r, 0] as ut(θ) = u(t+θ). This problem can be set in an abstract framework by setting H=L2(Ω) and V=H1 0(Ω),which are Hilbert spaces for the usual inner products and associated norms given by (u, v) = ZΩ u(x)v(x)dx, |u|= (ZΩ|u(x)|2dx)1/2, u, v ∈H, ((u, v)) = n X i=1 (Diu, Div),||u|| = ( n X i=1 ZΩ|Diu|2dx)1/2, u, v ∈V. Due to the continuous and dense injection V⊂H, identifying Hwith its dual H0, and denoting by A:V−→ V0the unbounded linear operator given by hAu, vi= ((u, v)) , u, v ∈V, it follows that Au =−∆ufor any u∈D(A),where D(A) = {u∈V:Au ∈H}=H1 0(Ω) TH2(Ω). We denote by λ1the first eigenvalue of A. Recall that D(A) is also a Hilbert space for the norm kukD(A)=|Au|, u ∈D(A). Our problem can be written as a second order differential equation in H: (2)    u00 +αu0+Au =f+h(t, ut), t > 0, u(t) = φ(t−τ), t ∈[τ−r, τ], u0(t) = φ0(t−τ), t ∈[τ−r, τ]. In general, if (X, k·kX) is a Banach space, we denote by CXthe space C0([−r, 0]; X) with the sup-norm, i.e., kφkCX= supθ∈[−r,0] kφ(θ)kX,for φ∈CX. Given another Banach space (Y, k·kY) such that the injection X⊂Yis continuous, we denote by CX,Y the Banach space CX∩C1([−r, 0]; Y) with the norm k·kCX,Y defined by kφk2 CX,Y =kφk2 CX+kφ0k2 CY,for φ∈CX,Y . We will use the spaces CD(A), CV, CH,CV,H ,and CD(A),V in our analysis. We make the following hypotheses on the function h:R×CH→H. (I) ∀ξ∈CH,t∈R→h(t, ξ)∈His continuous, (II) ∀t∈R,h(t, 0) = 0, (III) ∃Lh>0 such that ∀t∈R,∀ξ, η ∈CH |h(t, ξ)−h(t, η)| ≤ Lhkξ−ηkCH, ATTRACTORS FOR A DAMPED WAVE EQUATION 3 (IV) ∃m0≥0, Ch>0 such that ∀m∈[0, m0], τ ≤t, u, v ∈C0([τ−r, t]; H) Zt τ ems |h(s, us)−h(s, vs)|2ds≤C2 hZt τ−r ems |u(s)−v(s)|2ds. (V) h∈C1(R×CH;H),and there exists C > 0 such that, for any (t, ξ)∈R×CHthe Fr´echet derivative δh(t, ξ)∈ L(R×CH, H) satisfies ||δh(t, ξ)||L(R×CH,H)≤C(1 + ||ξ||CH). Notice that condition (V) implies (I). However, we prefer to state both assumptions since some of our results hold true by assuming only conditions (I)-(IV). We have the following result. Theorem 2.1. Assume that f∈L2 loc(R, H), φ ∈CV,H and hsatisfies (I)-(IV). Then, for any τ∈R, there exists a unique solution u(·) = u(·;τ, φ)of the problem (2) such that u∈ C0([τ−r, ∞); V)∩C1([τ−r, ∞); H).If in addition f0∈L2(τ, T;H)for all T > 0,φ(0) ∈CD(A), φ0(0) ∈CV, then u∈C0([τ, ∞); D(A)) ∩C1([τ, ∞); V). Proof. The existence is a consequence of a more general situation in [10] (see also [11] for a stochastic version). The regularity assertion follows from [14] (see also Appendix). ¤ 2.2. Preliminaries on pullback attractors. As the autonomous theory of attractors is not suitable to handle our model, we will use the theory of pullback attractors instead. In the case of nonautonomous differential equations the initial time is just as important as the actual time, and the classical semigroup property of autonomous dynamical systems is no longer available. This is why the recent theory of pullback attractors has proven very useful, even for random dynamical systems (see [12], [13], [8]). Instead of a family of one-parameter maps S(t) we need to use a two-parameter semigroup or process U(t, τ) on the complete metric space X(which in our case will be CV,H or CD(A),V ) (cf. Sell [16]); U(t, τ)ψdenotes the value of the solution at time twhich was equal to the initial value ψat time τ. The semigroup property is replaced by the process composition property (3) U(t, τ)U(τ, r) = U(t, r) for all t≥τ≥r, and, obviously, the initial condition implies U(τ, τ) =Id. As in the standard theory of attractors, we seek an invariant attracting set. However, since the equation is nonautonomous this set also depends on time. Definition 2.2. Let Ube a two-parameter semigroup on a complete metric space X. A family of compact sets {A(t)}t∈Ris said to be a (global) pullback attractor for Uif, for all τ∈R, it satisfies (i) U(t, τ)A(τ) = A(t)for all t≥τ, and (ii) lims→∞ distX(U(t, t −s)D, A(t)) = 0, for all bounded D⊂X, and all t∈R. The pullback attractor is said to be uniform if the attraction property is uniform in time, i.e. lim s→∞sup t∈R distX(U(t, t −s)D, A(t)) = 0,for all bounded D⊂X. In the definitions, distX(A, B) is the Hausdorff semidistance between Aand B, defined as distX(A, B) = sup a∈A inf b∈BdX(a, b),for A, B ⊆X. The notion of an attractor is closely related to those of pullback absorbing and attracting sets. 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Definition 2.3. The family {B(t)}t∈Ris said to be (a) pullback absorbing with respect to the process U, if for all t∈Rand all bounded D⊂X, there exists TD(t)>0such that for all s≥TD(t) U(t, t −s)D⊂B(t); (b) pullback attracting with respect to the process U, if for all t∈R, all bounded D⊂Xand all ε > 0,there exists Tε,D(t)>0such that for all s≥Tε,D(t) distX(U(t, t −s)D, B(t)) < ε; (c) pullback uniformly absorbing (resp. uniformly attracting) if TD(t)in part (a) (resp. Tε,D(t) in part (b)) does not depend on the time t. Related to these concepts we also have the notion of forward attracting sets. Definition 2.4. A family of sets {B(t)}t∈Ris said to be forward attracting for Uif, for all τ∈R, it satisfies limt→∞ distX(U(t, τ)D, B(t)) = 0, for all bounded D⊂X. The forward attraction is said to be uniform if lim t→∞sup τ∈R distX(U(t+τ, τ)D, B(t+τ)) = 0,for all bounded D⊂X. Remark 2.5. The previous definitions make also sense even if the family of operators U(·,·) does not satisfy the properties of a two-parameter semigroup. The existence of compact absorbing or attracting sets is crucial in order to prove the existence of pullback attractors. Due to the hyperbolicity of the equation, we will need the existence of compact attracting sets instead of absorbing ones. In the next theorem we collect some sufficient conditions ensuring these facts (see Crauel and Flandoli [7], Schmalfuss [15] or Crauel et al. [8]). Theorem 2.6. Let U(t, τ)be a two-parameter process, and suppose U(t, τ) : X→Xis continuous for all t≥τ. If there exists a family of compact pullback attracting sets {B(t)}t∈R, then there exists a pullback attractor {A(t)}t∈R, such that A(t)⊂B(t)for all t∈R, and which is given by A(t) = [ D⊂X bounded ΛD(t),where ΛD(t) = \ n∈N[ s≥n U(t, t −s)D. Remark 2.7. The uniqueness of the pullback attractor, as defined above, does not hold in general (see [1]). However, the one given in the preceding theorem is minimal with respect to set inclusion (see Crauel and Flandoli [7]). If we wish to guarantee uniqueness, we need to assume some additional hypotheses on the attractor, for instance, that the attractor belongs to a certain class of set valued functions (also called a universe), which are attracted by the attractor (see, e.g., Cheban et al. [4]). In particular, we may consider as a universe of attracted sets the family of uniformly bounded sets, which is equivalent to assume that the family {A(t)}t∈Ris uniformly bounded (i.e there exists a bounded set B⊂Xsuch that A(t)⊂Bfor all t∈R). A sufficient condition ensuring this is that the family of compact attracting sets in Theorem 2.6 is also uniformly bounded. Remark 2.8. Notice that, when the attraction of {B(t)}t∈Rin Theorem 2.6 is uniform, the attractor need not be. On the other hand, if that family is given by a single set, i.e. B(t) = B, for all t∈R, and the pullback attraction of this family is uniform, then this compact set Bis also uniformly forward attracting and, by means of the theory developed by Chepyzhov and Vishik [5], ATTRACTORS FOR A DAMPED WAVE EQUATION 5 there exists a uniform forward attractor Afwhich contains all the component sets of the pullback attractor. Moreover, it holds [ t∈RA(t)⊂Af, being the inclusion strict in general (see Chepyzhov and Vishik [5] for more details). The following result will be useful in order to prove the existence of the attracting family of sets. Proposition 2.9. Let Ube a process in the normed linear space X, and suppose that Ucan be written as U(t, τ) = U1(t, τ) + U2(t, τ)for all t≥τ, and let {B(t)}t∈Rbe a family of subsets of Xsuch that lim s→∞distX(U1(t, t −s)D, B(t)) = 0,for all t∈R,∀D⊂Xbounded (resp. lim s→∞sup t∈R distX(U1(t, t −s)D, B(t)) = 0,∀D⊂Xbounded), and lim s→∞µsup y∈D||U2(t, t −s)y||X¶= 0,for all t∈R,∀D⊂Xbounded (resp. lim s→∞sup t∈Rµsup y∈D||U2(t, t −s)y||X¶= 0,∀D⊂Xbounded). Then, lim s→∞distX(U(t, t −s)D, B(t)) = 0,for all t∈R,∀D⊂Xbounded (resp. lim s→∞sup t∈R distX(U(t, t −s)D, B(t)) = 0,∀D⊂Xbounded). 3. Existence of pullback and forward attractors We will proceed as follows. First, we will construct the evolution process associated to our problem. Then, we will prove the existence of a compact set which is uniformly attracting. As a consequence, the existence of a pullback attractor can be ensured as well as a forward uniform one. 3.1. The evolution process associated to the model. Consider our model (2) under the assumptions in Theorem 2.1 but, for simplicity, we assume that f∈H(we will comment on the time dependent situation later on). We can construct a two-parameter semigroup or process U(·,·) in CV,H as the family of mappings defined by U(t, τ)(φ) = ut(·;τ, φ)∀t≥τ, φ ∈CV,H . Observe that when φ∈CD(A),V , Theorem 2.1 implies that ut(·;τ, φ)∈CD(A),V ,which means that we also have a process defined in the space CD(A),V . From now on we will argue only for the first case. The evolution property (3) for the family {U(t, τ) : t≥τ}follows from the uniqueness of solutions established in Theorem 2.1. The continuity of each operator U(t, τ) is a consequence of the following result. Lemma 3.1. Let φ,ψ∈CV,H be two initial data for our problem (2), and let τ∈Rbe an initial time. Denote by u(·) = u(·;τ, φ)and v(·) = u(·;τ, ψ)the corresponding solutions to (2). Then, there exists a constant γ > 0which does not depend on the initial data and time, such that (4) ||u(t)−v(t)||2+|u0(t)−v0(t)|2≤¡1 + λ−1 1C2 hr¢eγ(t−τ)||φ−ψ||2 CV,H ∀t≥τ, and (5) ||ut−vt||2 CV,H ≤¡1 + λ−1 1C2 hr¢eγ(t−τ)||φ−ψ||2 CV,H ∀t≥τ+r. 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Proof. Let us denote w(·) = u(·)−v(·).Then, we have w00 +αw0+Aw =h(t, ut)−h(t, vt), 1 2 d dt(||w||2+|w0|2) + α|w0|2= (h(t, ut)−h(t, vt), w0) ≤ |h(t, ut)−h(t, vt)||w0| ≤1 2|w0|2+1 2|h(t, ut)−h(t, vt)|2, and thus (6) d dt(||w||2+|w0|2)≤ |w0|2+|h(t, ut)−h(t, vt)|2. Taking into account condition (IV) Zt τ|h(s, us)−h(s, vs)|2ds ≤C2 hZt τ−r|u(s)−v(s)|2ds ≤λ−1 1C2 hr||φ−ψ||2 CV+λ−1 1C2 hZt τ||w(s)||2ds, from (6) we obtain by integrating over the interval [τ, t] ||w(t)||2+|w0(t)|2≤ ||w(τ)||2+|w0(τ)|2+λ−1 1C2 hr||φ−ψ||2 CV +Zt τ¡λ−1 1C2 h||w(s)||2+|w0(s)|2¢ds ≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H +Zt τ¡λ−1 1C2 h||w(s)||2+|w0(s)|2¢ds, and, consequently, ||w(t)||2+|w0(t)|2≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H +γZt τ¡||w(s)||2+|w0(s)|2¢ds, where γ= max{1, λ−1 1C2 h}.The Gronwall lemma implies now ||w(t)||2+|w0(t)|2≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ)for all t≥τ. If we consider t≥τ+r, then t+θ≥τfor any θ∈[−r, 0] and ||w(t+θ)||2+|w0(t+θ)|2≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ+θ) ≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ). Thus, ||wt||2 CV,H ≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ),for t≥τ+r. ¤ Theorem 3.2. The mapping U(t, τ) : CV,H →CV,H is continuous for any t≥τ. Proof. Let φ,ψ∈CV,H be two initial data for our problem (2), and let t≥τ. Denote by u(·) = u(·;τ, φ) and v(·) = u(·;τ, ψ) the corresponding solutions to (2). Then, writing again w=u−vwe obtain the following. If τ−r≤t≤τ, then w(t) = φ(t−τ)−ψ(t−τ) and ||w(t)||2+|w0(t)|2≤ ||φ−ψ||2 CV+||φ0−ψ0||2 CH ≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ+r). ATTRACTORS FOR A DAMPED WAVE EQUATION 7 Thus we have ||w(t)||2+|w0(t)|2≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ+r)for t≥τ−r, whence ||wt||2 CV,H ≤¡1 + λ−1 1C2 hr¢||φ−ψ||2 CV,H eγ(t−τ+r)for t≥τ, which implies the continuity of U(t, τ) in its state variable. ¤ 3.2. Existence of bounded absorbing sets in CV,H .We will consider in our computations an equivalent norm in the space CV,H .For each ε∈R, the norm ||φ||2 ε=||φ||2 CV+||φ0+εφ||2 CH, φ∈CV,H,is equivalent to k·k0:= k·kCV,H . This allows us to obtain absorbing balls for the original norm by proving the existence of absorbing balls for this new norm for some suitable value of ε. Indeed, let us denote Bε(0, ρ) = {φ∈CV,H :||φ||ε< ρ}. Noticing that for c1= max{2,1+2ε2λ−1 1}it follows that ||φ||2 CV,H =||φ||2 CV+||φ0+εφ −εφ||2 CH ≤ ||φ||2 CV+ 2||φ0+εφ||2 CH+ 2ε2||φ||2 CH ≤¡1+2ε2λ−1 1¢||φ||2 CV+ 2||φ0+εφ||2 CH ≤c1||φ||2 ε, we then have Bε(0, ρ)⊂B0³0, c1/2 1ρ´. Theorem 3.3. Assume that Conditions (I)-(IV) hold with m0>0, and that f∈H. Suppose in addition that 2√2λ−1/2 1Ch<min{α 4,λ1 2α}.Then, there exists a family {B(t)}t∈Rof bounded sets in CV,H which is uniformly pullback (and forward) absorbing for the process U(·,·).Moreover, B(t) = B0for all t∈R, where B0is a bounded set in CV,H . Proof. We first choose ε > 0 such that 2√2λ−1/2 1Ch< ε < min{α 4,λ1 2α}.Let D⊂CV,H be a bounded set, i.e. there exists d > 0 such that for any φ∈Dit holds ||φ||2 ε=||φ||2 CV+||φ0+εφ||2 CH≤d2. Now denote by u(·) = u(·;τ, φ) the solution of problem (2). Writing v=u0+εu and taking into account equality (28) in the Appendix, we have 1 2 d dt(|v|2+||u||2) + ε||u||2+ (α−ε)|v|2−ε(α−ε)(u, v) = (f, v)+(h(t, ut), v). Now, for α1=ε 2,we obtain ε||u||2+ (α−ε)|v|2−ε(α−ε)(u, v)≥ε||u||2+ (α−ε)|v|2−ε(α−ε) √λ1||u|||v| ≥ε||u||2+3α 4|v|2−εα √λ1||u|||v| ≥ε||u||2+3α 4|v|2−ε 2||u||2−εα2 2λ1|v|2 ≥ε 2||u||2+α 2|v|2 ≥α1(||u||2+|v|2). 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Thus, d dt(||u||2+|v|2)+2α1(||u||2+|v|2)≤2|f||v|+ 2|h(t, ut)||v| ≤α1 2|v|2+α1 2|v|2+2 α1|h(t, ut)|2+2 α1|f|2 ≤α1|v|2+2 α1|h(t, ut)|2+2 α1|f|2, whence d dt (||u||2+||v||2) + α1(||u||2+|v|2)≤2 α1|f|2+2 α1|h(t, ut)|2. As our assumptions ensure that −α1+2λ−1 1C2 h α1<0,we can then choose m∈(0, m0) small enough such that m−α1+2λ−1 1C2 h α1<0.For this choice, we have d dt £emt ¡||u||2+|v|2¢¤ =memt ¡||u||2+|v|2¢+emt d dt £||u||2+|v|2¤ and d dt £emt ¡||u||2+|v|2¢¤ ≤(m−α1)emt ¡||u||2+|v|2¢+2emt α1|f|2+2emt α1|h(t, ut)|2. By integrating over the interval [τ, t],we deduce emt ¡||u(t)||2+|v(t)|2¢≤emτ ¡||u(τ)||2+|v(τ)|2¢+Zt τ 2ems α1|f|2ds + (m−α1)Zt τ ems ¡||u(s)||2+|v(s)|2¢ds +Zt τ 2ems α1|h(t, ut)|2ds ≤emτ ¡||u(τ)||2+|v(τ)|2¢+2 α1m|f|2(emt −emτ ) + (m−α1)Zt τ ems ¡||u(s)||2+|v(s)|2¢ds +2C2 hλ−1 1 α1Zt τ−r ems||u(s)||2ds =emτ ¡||u(τ)||2+|v(τ)|2¢+2 α1m|f|2(emt −emτ ) + (m−α1)Zt τ ems ¡||u(s)||2+|v(s)|2¢ds +2C2 hλ−1 1 α1·Zτ τ−r ems||φ(s)||2ds +Zt τ ems||u(s)||2ds¸ ≤emτ d2³1 + 2λ−1 1C2 hr α1´+2 α1m|f|2(emt −emτ ) +³m−α1+2λ−1 1C2 h α1´Zt τ ems ¡||u(s)||2+|v(s)|2¢ds ≤emτ d2³1 + 2λ−1 1C2 hr α1´+2 α1m|f|2(emt −emτ ), and this implies, for t≥τ, ||u(t)||2+|v(t)|2≤emτ e−mtd2µ1 + 2λ−1 1C2 hr α1¶+2 α1m|f|2¡1−emτ e−mt¢ ≤2 α1m|f|2+emτ e−mtd2µ1 + 2λ−1 1C2 hr α1¶.(7) ATTRACTORS FOR A DAMPED WAVE EQUATION 9 If we denote ρ2 0=4c1 α1m|f|2,ˆρ2 0=c1µ1 + 2λ−1 1C2 hr α1¶, then (7) yields that (8) ||u(t;τ, φ)||2+|u0(t;τ, φ)|2≤ρ2 0 2+ ˆρ2 0d2em(τ−t),∀t≥τ, and, in particular, (9) ||u(t;τ, φ)||2+|u0(t;τ, φ)|2≤ρ2 0 2+ ˆρ2 0d2,∀t≥τ. Moreover, as u(t;τ, φ) = φ(t−τ) and u0(t;τ, φ) = φ0(t−τ) for t∈[τ−r, τ],then Eq. (9) holds true for t≥τ−r. If we take now t≥τ+r, then for all θ∈[−r, 0] we have t+θ≥τand so (10) ||u(t+θ;τ, φ)||2+|u0(t+θ;τ, φ)|2≤ρ2 0 2+ ˆρ2 0d2emrem(τ−t), or, in other words, (11) ||U(t, τ)φ||2 CV,H ≤ρ2 0 2+ ˆρ2 0d2emrem(τ−t),∀t≥τ+r, φ ∈D. Therefore, there exists TD≥rsuch that (12) ||U(t, t −s)φ||2 CV,H ≤ρ2 0,∀t∈R, s ≥TD, φ ∈D, which means that the ball BCV,H (0, ρ0) = B0⊂CV,H is uniformly pullback absorbing for the process U(·,·) (but it is not compact yet). ¤ Remark 3.4. On the one hand, observe that if t0∈Rand t≥t0,then u(t+θ;t0−s, φ) = u(t+θ;t−(s+t−t0), φ)and u0(t+θ;t0−s, φ) = u0(t+θ;t−(s+t−t0), φ)with s+t−t0≥s. As a consequence of (12) we have ||U(t, t0−s)φ||2 CV,H ≤ρ2 0,∀t0∈R, t ≥t0, s ≥TD, φ ∈D, or equivalently, we have ∀t0∈R, t ≥t0, θ ∈[−r, 0], s ≥TD, φ ∈D, (13) ||u(t+θ;t0−s, φ)||2+|u0(t+θ;t0−s, φ)|2≤ρ2 0. On the other hand, Eq. (9) implies, ∀t0∈R, s ∈R, t ≥t0−s−r, φ ∈D, (14) ||u(t;t0−s, φ)||2+|u0(t;t0−s, φ)|2≤ρ2 0 2+ ˆρ2 0d2. 3.3. Existence of attracting sets in CV,H .Now we can prove the following result. Theorem 3.5. In addition to the assumptions in Theorem 3.3, suppose that Condition (V) holds. Then, there exists a compact set B2⊂CV,H which is uniformly pullback attracting for the process U(·,·),and consequently, there exists the pullback attractor {A(t)}t∈R, and the uniform forward one (in the sense of Chepyzhov and Vishik) Af⊂CV,H . Moreover, A(t)⊂CD(A),V for all t∈R. Proof. Let us fix εas in the proof of Theorem 3.3. Let D⊂CV,H be a bounded set, i.e. there exists d > 0 such that for any φ∈Dit holds ||φ||2 ε≤d2,and so, ||φ||2 CV,H ≤c1d2. 16 TOM´ AS CARABALLO, PETER E. 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Sell, Non-autonomous differential equations and dynamical systems, Amer. Math. Soc. 127(1967), 241-283. [17] W.A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19(1966), No. 3, 543-551. [18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, (1988). E-mail address, Tom´as Caraballo: [email protected] E-mail address, Peter E. Kloeden: [email protected] E-mail address, Jos´e Real: [email protected] ATTRACTORS FOR A DAMPED WAVE EQUATION 17 (Tom´as Caraballo and Jos´e Real) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain) (Peter E. Kloeden) Fachbereich Mathematik, Johann Wolfgang Goethe-Universit¨ at, D-60054 Frankfurt am Main, Germany