PULLBACK EXPONENTIAL ATTRACTORS FOR EVOLUTION PROCESSES IN BANACH SPACES: PROPERTIES AND APPLICATIONS

. This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.


1.
Introduction. Global pullback attractors proved to be a useful tool to study the asymptotic dynamics of infinite dimensional non-autonomous dynamical systems.
To be more precise, let here and in the sequel (X, d X ) denote a complete metric space and T = R or T = Z. The rules of time evolution in the non-autonomous setting are dictated by a two-parameter family U = {U (t, s)| t ≥ s}, t, s ∈ T, of continuous operators from X into itself, which is called an evolution process in X, if it satisfies the properties ∀ t ≥ s, If an evolution process possesses a pullback exponential attractor {M(t)| t ∈ T}, the existence of the global pullback attractor {A(t)| t ∈ T} follows immediately from Theorem 1.2. Moreover, the global pullback attractor is contained in the pullback exponential attractor and possesses finite dimensional sections. Indeed, the minimality property in Definition 1.1 implies An algorithm for the construction of non-autonomous exponential attractors was first developed in [12] for discrete evolution processes, where the authors considered forwards exponential attractors. The method is based on the compact embedding of the phase space V into an auxiliary normed space W and the smoothing or regularizing property of the evolution process (see Section 2). Using the pullback approach the result was recently extended in [8] and [16] for time continuous evolution processes. Common assumptions in both articles were that the process satisfies the smoothing property, which implies that it is eventually compact, and the existence of a fixed bounded uniformly pullback absorbing set. This allows the pullback exponential attractor M to be unbounded in the future but it is always uniformly bounded in the past, i.e., the union t≤t0

M(t)
is bounded for all t 0 ∈ T. Moreover, the Hölder continuity in time of the evolution process was essential for the construction in [8] and [16]. It is typically satisfied in parabolic problems, but not by evolution processes generated by hyperbolic equations. We proposed an alternative method for time-continuous evolution processes in [5], which does not require the Hölder continuity in time of the evolution process, we extended the algorithm for evolution processes that are asymptotically compact and considered a time-dependent family of bounded pullback absorbing sets instead of a fixed bounded pullback absorbing set. Our construction leads to better bounds for the fractal dimension of the sections of the attractors and to existence results for pullback exponential attractors that are not necessarily uniformly bounded in the past. To prove the finite fractal dimension of global pullback attractors that are not uniformly bounded in the past has been an open problem. Previous constructions of pullback exponential attractors were therefore limited to evolution processes possessing global pullback attractors that are uniformly bounded in the past (see Section 1 in [16] and Remark 3.2 in [18]).
In [13] the authors proposed a construction for forwards exponential attractors for time continuous evolution processes, which is similar to our method. However, the existence of the uniform attractor for the evolution process is a priori known and the existence of a fixed bounded uniformly forwards absorbing set is assumed. This is equivalent to the assumption of a fixed bounded uniformly pullback absorbing set and implies the uniform boundedness of the forwards exponential attractor (i.e., t∈T M(t) is bounded). They consider asymptotically compact evolution processes in the weaker space W , a construction for processes that are asymptotically compact in the stronger phase space V as we formulated in [5] has not been considered before (see [8], [16], [13] and for autonomous exponential attractors [11], [12]). We discussed and compared these different settings and results in [5], Section 3.2.
Our present article is the continuation of [5], where we constructed pullback exponential attractors for asymptotically compact evolution processes in Banach spaces assuming that the process possesses a family of time-dependent pullback absorbing sets that possibly grow in the past and studied its implications in the autonomous setting. We now discuss properties of the attractors and apply the theoretical results to prove the existence of pullback exponential attractors in two applications.
In both examples, previous results are not applicable and the generalizations we developed in [5] are essentially needed.
In particular, we consider a non-autonomous Chafee-Infante equation in a bounded domain Ω ⊂ R n , n ∈ N, where λ > 0 and the initial data u s ∈ C(Ω). The non-autonomous term β : R → R + is strictly positive, continuously differentiable, bounded when time t tends to ∞ and vanishes as t goes to −∞. We show that the generated evolution process satisfies the smoothing property and possesses a semi-invariant family of pullback absorbing sets. The diameter of the absorbing sets grows in the past since the function β vanishes when t tends to −∞. From our results in [5] we deduce the existence of a pullback exponential attractor for the generated evolution process. This implies that the global pullback attractor exists and that its sections are of finite fractal dimension. Furthermore, we prove that the global pullback attractor is unbounded in the past, lim where diam denotes the diameter in the space C(Ω), which provides a positive answer to the question whether the finite fractal dimension can be established for global pullback attractors that are not uniformly bounded in the past (see Section 1 in [16] and Remark 3.2 in [18]).
The second application is the non-autonomous dissipative wave equation where Ω ⊂ R n , n ∈ N, n ≥ 3, is a bounded domain. We assume that the nonlinearity f : R → R is continuously differentiable and of sub-critical growth. The initial value problem generates an asymptotically compact evolution process U in the phase space V := H 1 0 (Ω) × L 2 (Ω). We prove that the evolution process can be represented as a sum U = S + C, where the family of operators S satisfies the smoothing property with respect to V and an auxiliary normed space W compactly embedded into V , and C is a family of contractions in the stronger space V . Our main result in [5] implies the existence of a pullback exponential attractor for the evolution process U . Previous results cannot be applied since the constructions of exponential attractors were developed for evolution processes or semigroups that are asymptotically compact in the weaker space W , i.e., under the assumption that the family C is a contraction in W (among others see [11], [12] and [13]). Moreover, the former existence results for pullback exponential attractors in [8] and [16] required the Hölder continuity in time of the evolution process, which is generally not satisfied by hyperbolic equations.
The outline of our paper is as follows. In Section 2 we recall the main result of [5] about the existence of pullback exponential attractors for asymptotically compact evolution processes. We discuss properties of the pullback exponential attractors and consequences of our existence theorem in Section 3. Finally, in Section 4 we apply our theoretical results and show the existence of pullback exponential attractors for a non-autonomous damped wave equation and a non-autonomous Chafee-Infante equation.

2.
A general existence theorem for pullback exponential attractors. In this section we recall the existence result for pullback exponential attractors obtained in [5]. Let U = {U (t, s)| t ≥ s} be an evolution process in the Banach space (V, · V ). The construction of the pullback exponential attractor is based on the existence of a time-dependent pullback absorbing family, the compact embedding of the phase space into an auxiliary normed space and the asymptotic smoothing property of the process. We assume the process U can be represented as U = S +C, where {S(t, s)| t ≥ s} and {C(t, s)| t ≥ s} are families of operators satisfying the following properties: for some constant µ > 0. (H 1 ) There exists a family of bounded sets B(t) ⊂ V , t ∈ T, that pullback absorbs all bounded subsets of V : For every bounded set D ⊂ V and every t ∈ T there exists a pullback absorbing time T D,t ∈ T + := {t ∈ T| t ≥ 0} such that (H 2 ) The family {S(t, s)| t ≥ s} satisfies the smoothing property within the absorbing sets: There existt ∈ T + \{0} and a constant κ > 0 such that (H 3 ) The family {C(t, s)| t ≥ s} is a contraction within the absorbing sets: where the contraction constant 0 ≤ λ < 1 2 . (H 4 ) The process {U (t, s)| t ≥ s} is Lipschitz continuous within the absorbing sets: For all t ∈ T and t ≤ s ≤ t +t there exists a constant L t,s > 0 such that The construction of pullback exponential attractors requires to impose additional assumptions on the pullback absorbing family in Hypothesis (H 1 ).
(A 1 ) The family of absorbing sets {B(t)| t ∈ T} is positively semi-invariant for the evolution process {U (t, s)| t ≥ s}, (A 2 ) For every bounded subset D ⊂ V and time t ∈ T the corresponding absorbing times are bounded in the past: There exists T D,t ∈ T + such that The above-stated assumptions allow to construct pullback exponential attractors for the evolution process {U (t, s)| t ≥ s} (see [5]). In the sequel, we denote by B X r (a) the ball of radius r > 0 and center a ∈ X in the metric space X and by N X ε (A) the minimal number of balls in X with radius ε > 0 and centers in A needed to cover the subset A ⊂ X.
Theorem 2.2. Let {U (t, s)| t ≥ s}, t, s ∈ T, be an evolution process in the Banach space V and the assumptions (H 0 )-(H 4 ), (A 1 ) and (A 2 ) be satisfied. Moreover, we assume that the diameter of the family of absorbing sets {B(t)| t ∈ T} grows at most sub-exponentially in the past. Then, for every ν ∈ (0, 1 2 − λ) there exists a pullback exponential attractor {M ν (t)| t ∈ T} for the evolution process {U (t, s)| t ≥ s}, and the fractal dimension of its sections is uniformly bounded by Remark 1. For discrete evolution processes the Lipschitz continuity assumption (H 4 ) in Theorem 2.2 can be omitted.

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3. Properties of the pullback exponential attractor. An immediate consequence of Theorem 2.2 is the existence and finite dimensionality of the global pullback attractor. For the proof of the following theorem we define the group of time shift operators or temporal translations {S r | r ∈ T} by where r ∈ T and {U (t, s)| t ≥ s} is an evolution process.
be an evolution process in the Banach space V and the assumptions (H 0 )-(H 3 ), (A 1 ) and (A 2 ) be satisfied. Moreover, we assume that the diameter of the family of absorbing sets {B(t)| t ∈ T} grows at most sub-exponentially in the past. Then, the global pullback attractor {A(t)| t ∈ T} of the evolution process {U (t, s)| t ≥ s} exists, and the fractal dimension of its sections is uniformly bounded by Proof. For discrete evolution processes the statements follow from Theorem 2.2, Remark 1 and the minimality property of the global pullback attractor (see Definition It satisfies the assumptions of Theorem 2.2, and we conclude that for every ν ∈ (0, 1 2 − λ) there exists a pullback exponential attractor {M ν d (k)| k ∈ Z} for the discrete evolution process { U (n, m)| n ≥ m}. We define the sets , | t ∈ R} is a family of compact subsets of V . Moreover, it follows as in the proof of Theorem 2.2 that the family { M ν (t)| t ∈ R} pullback attracts all bounded subsets of V . By Theorem 1.2 we conclude that the global pullback attractor {A(t)| t ∈ R} of the time continuous process {U (t, s)| t ≥ s} exists, and the minimality property implies Since ν ∈ (0, 1 2 − λ) was arbitrary Theorem 2.2 implies that the fractal dimension of the discrete global pullback attractor is uniformly bounded by and it remains to estimate the fractal dimension of the time continuous sections. To this end let r ∈ R be arbitrary. We consider the shifted evolution process {S r U (t, s)| t ≥ s} and the associated discrete evolution process { U r (n, m)| n ≥ m}, which is given by U r (n, m) := U r (nt, mt) for all n ≥ m, n, m ∈ Z. By Theorem 2.2 and Remark 1 for every ν ∈ (0, 1 2 − λ) there exists a pullback exponential attractor {M ν r,d (k)| k ∈ Z} for the discrete evolution process { U r (n, m)| n ≥ m}, and the fractal dimension of its sections satisfies the estimate stated in the theorem. We follow the previous arguments to conclude that the global pullback attractor {A r (t)| t ∈ R} for the time continuous evolution process {S r U (t, s)| t ≥ s} exists and observe that Moreover, the fractal dimension of the discrete sections of the global pullback attractor is uniformly bounded, for all k ∈ Z. Finally, since r ∈ R was arbitrary and we obtain the uniform bound for the fractal dimension of the time continuous global pullback attractor {A(t)| t ∈ R}.
Remark 2. We remark that the Lipschitz continuity (H 4 ), which is essential for the construction of the time continuous pullback exponential attractor, is not required to establish the existence of the global pullback attractor and to derive estimates on its fractal dimension (see the hypothesis in Theorem 3.1).
) and was introduced by Kolmogorov and Tihomirov in [15]. The order of growth of H ε as ε tends to zero is a measure for the massiveness of the set A in X, even if the fractal dimension of A is infinite.
The bound on the fractal dimension of the global pullback attractor in Theorem 3.1 is related to the entropy numbers for the embedding of the spaces V and W . For k ∈ N the k-th entropy number for the embedding V → W is defined as If V and W are infinite dimensional Banach spaces such that the embedding V → → W is compact, then 0 < e k < ∞ for all k ∈ N. Let λ = 0 and ν ∈ (0, 1 2 ) be as in Theorem 3.1, that is, the evolution process U satisfies the smoothing property. Assuming that e k → 0 as k → ∞ and that there exists k ∈ N such that e k = ν κ we obtain in our estimate .
We further observe that On the other hand, if the entropy numbers grow polynomially in 1 k , i.e., if e k = c k α for some constants c, α > 0, then and consequently,

PULLBACK EXPONENTIAL ATTRACTORS: PROPERTIES AND APPLICATIONS 1149
These observations illustrate that there exists an optimal constant ν ∈ (0, 1 2 ) to minimize the bound on the fractal dimension in Theorem 3.1.
For certain function spaces the entropy numbers can explicitly be estimated (see [10]). For instance, for the embeddings of the Sobolev spaces W s1,p (Ω) into W s2,q (Ω), where Ω ⊂ R n is a smooth bounded domain and s 1 , for some constants c 1 , c 2 ≥ 0 (Theorem 2, Section 3.3.3 in [10]), and our argumentation above applies.
The following proposition illustrates the relation between the global pullback and the pullback exponential attractor for evolution processes. We recall that an evolution process U was called B-asymptotically compact in [19], where B = {B(t)| t ∈ T} is a family of bounded subsets, if for every t ∈ T and all sequences Under the assumptions of Theorem 2.2 it can be observed from its proof in [5] that the evolution process U is Basymptotically compact, where the family B is the family of pullback absorbing sets B = {B(t)| t ∈ T} in Assumption (H 1 ). Moreover, it follows from [19] that Λ( B, t), t ∈ T, is a strictly invariant family of non-empty, compact subsets of V that pullback attracts all bounded sets and However, the sets do not coincide in general.

Remark 4.
For an evolution process {U (t, s)| t ≥ s} satisfying the hypotheses of Theorem 2.2 the pullback exponential attractor in [5] was defined as (see the proof of Theorem 3.2 and Theorem 3.3). The family of discrete sets E n (k), n ∈ N 0 , k ∈ Z, satisfies the properties (2(ν+λ)) n R k−n (u), where denotes the cardinality of a set, B(k) are the pullback absorbing sets for discrete times k ∈ Z in Hypothesis (H 1 ), and R k > 0 is the radius of a ball in V that contains B(k). Proposition 1. Let {U (t, s)| t ≥ s} be an evolution process in the Banach space V and the assumptions of Theorem 2.2 be satisfied. Then, the pullback exponential attractor of Theorem 2.2 can be represented as wheret is given by (H 2 ) and (H 3 ), and we refer to [5] for the definition and construction of the family of sets E n (k), n ∈ N 0 , k ∈ Z. Moreover, if the family of pullback absorbing sets is bounded in the past, i.e., if the union t≤t0 B(t) is bounded for all t 0 ∈ T, then where {A(t)| t ∈ T} denotes the global pullback attractor of the evolution process.
Proof. The pullback exponential attractor in [5] was defined as kt)E nm (k) and since the set is finite, Otherwise, there exists a subsequence, which we denote by (n m ) m∈N as well, such that lim m→∞ n m = ∞. By the definition of the sets E n (k) we have for some y m ∈ B (k − n m )t . It follows that x ∈ Λ( B, t), and we conclude To show the reverse inclusion let t ∈ T and x ∈ Λ( B, t). Then, there exist sequences (t m ) m∈N in T + , lim m→∞ t m = ∞, and (x m ) m∈N in B(t − t m ) such that x = lim m→∞ U (t, t − t m )x m . We argue by contradiction and assume that there exist ε > 0 and N 0 ∈ N such that and obtain by the definition of the pullback exponential attractor

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for some constant L ≥ 0, where we used the Lipschitz-continuity (H 1 ) in the second inequality and the semi-invariance of the absorbing sets in the last inequality. It follows from the proof of Theorem 2.2 in [5] that where the sequence of radii r km → 0 as k m tends to ∞. We conclude that if m ∈ N is sufficiently large, which contradicts our assumption and shows the relation Λ( B, t) ⊂ M ν (t).
To prove the second statement in the proposition it suffices to show the inclusion since the global pullback attractor is contained in the pullback exponential attractor. Let k ∈ Z, t ∈ [kt, (k + 1)t[, x ∈ M ν (t) and (x m ) m∈N be a sequence in n∈N0 U (t, kt)E n (k) such that lim m→∞ x m = x. For every m ∈ N there exists n m ∈ N 0 such that x m ∈ U (t, kt)E nm (k). If N 0 := sup{n m | m ∈ N} < ∞, it follows as above that Otherwise, there exists a subsequence, which we denote by (n m ) m∈N as well, such that lim m→∞ n m = ∞, and by the definition of the sets E n (k) we have x m = U t, (k − n m )t y m for some y m ∈ B((k−n m )t). By assumption, the family of absorbing sets is bounded in the past, which implies that {y m | m ∈ N} ⊂ s≤t B(s) ⊂ D for some bounded set D ⊂ V. It follows that where we used the representation of the global pullback attractor in Theorem 1.2.

Remark 5.
Let {U (t, s)| t ≥ s} be an evolution process in V, the hypothesis of Theorem 2.2 be satisfied and {A(t)| t ∈ T} and {M ν (t)| t ∈ T} be the corresponding global and exponential pullback attractor. We remark that is a countable dense subset of the section A(t) of the global pullback attractor for every t ∈ [kt, (k + 1)t[, k ∈ Z.
Moreover, if the pullback exponential attractor is bounded in the past, Proposition 1 implies that the Hausdorff dimensions of the sections A(t) and of M ν (t) coincide, dist V H (M ν (t)) = dist V H (A(t)) ∀t ∈ T, since the Hausdorff dimension of every countable set is zero. In this case, if we required finite Hausdorff instead of finite fractal dimension in the definition of exponential attractors we could add an arbitrarily large countable semi-invariant set to the global attractor without changing its dimension. This is not possible if we impose finite fractal dimension in the definition of exponential attractors (see also [9], Chapter 7, for the autonomous case).
If an evolution process {U (t, s)| t ≥ s} possesses the global pullback attractor {A(t)| t ∈ T} and is periodic, that is S r U = U for some r ∈ T, the invariance property U (t, s)A(s) = A(t) ∀ t ≥ s, t, s ∈ T, shows that the periodicity is directly inherited by the attractor. Since pullback exponential attractors are not unique we could certainly construct for an evolution process U and the shifted process S r U , where r ∈ T, pullback exponential attractors M U and M SrU that do not satisfy the cocycle property However, if {M U (t)| t ∈ T} is a pullback exponential attractor for the evolution process U the translation of the attractor {M U (t + r)| t ∈ T} yields a pullback exponential attractor for the shifted process S r U , for every r ∈ T.
Corollary 1. Let {U (t, s)| t ≥ s} be an evolution process in the Banach space V . We assume that the hypotheses of Theorem 2.2 are satisfied and denote by {M ν U (t)| t ∈ T} the pullback exponential attractor for {U (t, s)| t ≥ s} in Theorem 2.2. Then, for every r ∈ T the family {M ν is a pullback exponential attractor for the evolution process {S r U (t, s)| t ≥ s}, and the family of exponential attractors satisfies If an evolution process is periodic the family of pullback exponential attractors {M ν SrU (t)| t ∈ T} r∈T exhibits the same property. Proof. Let r ∈ T, {M ν U (t)| t ∈ T} be the pullback exponential attractor for the evolution process {U (t, s)| t ≥ s} in Theorem 2.2 and Then, the family {M ν SrU (t)| t ∈ T} is semi-invariant under the action of the evolution process {S r U (t, s)| t ≥ s}. The exponential pullback attraction property with respect to the process {S r U (t, s)| t ≥ s}, the compactness of the sections and the uniform bound for its fractal dimension immediately follow from the corresponding properties of the family {M ν U (t)| t ∈ T}, which proves that {M ν SrU (t)| t ∈ T} is a pullback exponential attractor for the shifted process.
Finally, we formulate assumptions for the construction of forwards exponential attractors.

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We replace the hypothesis (H 1 ) and (A 2 ) by the following: (H 1 ) There exists a family of bounded subsets B(t) ⊂ V , t ∈ T, that forwards absorbs all bounded subsets of V : For every bounded set D ⊂ V and every t ∈ T there exists a forwards absorbing time T D,t ∈ T + such that (A 2 ) For every bounded subset D ⊂ V and time t ∈ T the corresponding absorbing times are bounded in the future: There exists T D,t ∈ T + such that Then, for every ν ∈ (0, 1 2 − λ) there exists a forwards exponential attractor {M ν (t)| t ∈ T} for the evolution process {U (t, s)| t ≥ s}, and the fractal dimension of its sections is uniformly bounded by For discrete evolution processes Hypothesis (H 4 ) can be omitted.
Proof. Forwards exponential attractors can be constructed by slightly modifying the proof for pullback exponential attractors in [5].
Remark 6. If the pullback absorbing time T D,t corresponding to a bounded subset D ⊂ X in Hypothesis (H 1 ) is independent of the time t ∈ T, the family {B(t)| t ∈ T} is also forwards absorbing for the process. More precisely, the properties (H 1 ) and (H 1 ) are indeed equivalent in this case, and the conditions (A 2 ) and (A 2 ) are automatically satisfied. Consequently, in this case the pullback exponential attractor constructed in Theorem 2.2 coincides with the forwards exponential attractor in Theorem 3.3.

4.
Applications. In this section we illustrate our results and prove the existence of pullback exponential attractors for evolution processes generated by non-autonomous PDEs.

4.1.
Non-autonomous Chafee-Infante equation. The following initial value problem for the non-autonomous Chafee-Infante equation yields an example for a finite dimensional global pullback attractor which is unbounded in the past.
Let Ω ⊂ R n , n ∈ N, be a bounded domain with smooth boundary ∂Ω and s ∈ R. We consider the initial-/boundary value problem where the constant λ > 0, ∆ denotes the Laplace operator with respect to the spatial variable x, ∂ ∂ν the outward unit normal derivative on the boundary ∂Ω and ∂ ∂t the partial derivative with respect to time t > s. The initial data u s is a uniformly continuous function on Ω, u s ∈ C(Ω). Moreover, we assume that the non-autonomous term β : R → R + is strictly positive, continuously differentiable and satisfies the properties where the constants 0 < β 0 , β 1 < ∞. We consider the evolution process generated by (1) in the phase space W := C(Ω), where the norm in W is defined by To show the existence of a positively semi-invariant family of absorbing sets we use the method of lower and upper solutions (see [21], Chapter 2).
Since β vanishes slowly, which proves that c * is an upper solution for Problem (1).

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The non-linearity is odd with respect to u, and hence, we obtain Consequently, c * := −c * is a lower solution for (1) if the initial data satisfies u s (x) ≥ c * (s) for all x ∈ Ω.
The linear heat equation generates an analytic semigroup {e ∆t | t ∈ R + } in the space W := (C(Ω), · W ) (see [20]). We denote the associated fractional power spaces by X α , α ≥ 0. The operators e ∆t are linear and bounded from W to X α and satisfy the estimates where the constant C α ≥ 0 and · L(W ;X α ) denotes the operator norm. The semi-linear problem (1)  (see [20] and [22]). We apply Lemma 4.2 to show the existence of a semi-invariant family of pullback absorbing sets.

Proposition 2. The family of subsets
is positively semi-invariant for the evolution process {U (t, s)| t ≥ s} generated by Problem (1) and pullback absorbs all bounded sets of W .
Proof. Let s ∈ R and the initial data u s ∈ W satisfy u s W ≤ c * (s). Lemma 4.2 implies that the functions c * and c * are upper and lower solutions for Problem (1). From Theorem 4.1, Chapter 2, in [21] it follows that there exists a unique classical solution u( · , · ; u s , s) : Consequently, the evolution process {U (t, s)| t ≥ s} satisfies which proves the semi-invariance of the absorbing sets {B(t)| t ∈ R}.
To show that the family is pullback absorbing, let D ⊂ W be bounded and t ∈ R.
for all t ≤ t 0 , and consequently, D ⊂ B(t) for all t ≤ t 0 .
Finally, we observe that the pullback absorbing time is bounded in the past, in particular, T D,s ≤ t − t 0 for all s ≤ t.
Next, we show that {U (t, s)| t ≥ s} satisfies the smoothing property with respect to the Banach spaces and W , where the norm in V is defined by Proof. Let s ∈ R and u, v ∈ B(s) be given initial data. We denote the corresponding solutions of Problem (1) by u(t) := U (t, s)u and v(t) := U (t, s)v, t ≥ s. It was shown in [20], Theorem 2.4, that the continuous embedding X α → V exists for all α > 1 2 . Using the variation of constants formula we obtain where c ≥ 0 denotes the embedding constant. By Proposition 2 it follows that for some constant C ≥ 0, where we used Assumption (2) in the last estimate. The estimate (8) and the embedding V → W now imply

PULLBACK EXPONENTIAL ATTRACTORS: PROPERTIES AND APPLICATIONS 1157
for some constant µ > 0. Finally, we set t = s + 1 and and obtain the inequality Using the generalized Gronwall Lemma (Theorem 1.26 in [24]) we conclude for some constant κ > 0. Theorem 2.2 now implies the existence of a pullback exponential attractor in V for the evolution process {U (t, s)| t ≥ s}.
Remark 7. For evolution processes that satisfy the smoothing property it suffices to assume that the pullback absorbing family is bounded in the metric of W and that the process satisfies the Lipschitz continuity property (H 4 ) in W .
Indeed, if the family of absorbing sets is bounded in the metric of W we define the sets which are pullback absorbing and bounded in the space V by the smoothing property (H 2 ). Moreover, the smoothing property (H 2 ), the Lipschitz continuity in W and the continuous embedding (H 0 ) imply for all u, v ∈ B(t), t ∈ R and s ∈ [0,t]. This proves the Lipschitz continuity of the evolution process in the space V and the result remains valid.
Theorem 4.4. Let {U (t, s)| t ≥ s} be the evolution process in W = C(Ω) generated by Problem (1) and the function β satisfy Properties (2)- (5). Then, for every ν ∈ (0, 1 2 ) there exists a pullback exponential attractor {M ν (t)| t ∈ R} in V = C 1 (Ω) for the evolution process {U (t, s)| t ≥ s}, and the fractal dimension of its sections is uniformly bounded by where κ > 0 denotes the smoothing constant in Lemma 4.3. Furthermore, the global pullback attractor exists and is unbounded in the past, It is contained in the pullback exponential attractor, A(t) ⊂ M ν (t), and Proof. The family of pullback absorbing sets {B(t)| t ∈ R} defined in Lemma 4.3 satisfies the hypothesis (A 1 ) and (A 2 ). Since the diameter of the absorbing sets is bounded by and the non-autonomous term satisfies Property (5), the absorbing sets grow at most sub-exponentially in the past. Moreover, the embedding V → → W is compact, and the smoothing property with respect to the spaces V and W was shown in Lemma 4.3. To deduce the existence of a pullback exponential attractor from Theorem 2.2 it remains to verify the Lipschitz continuity of the evolution process. Let s ∈ R and u, v ∈ B(s) be given initial data. Using the variation of constants formula we obtain for some constant C 0 ≥ 0, where we used the estimate (9) y(s) = y 0 s ∈ R, y 0 ∈ R, also solve Problem (1) with initial data u s (x) = y 0 , x ∈ Ω. As shown in [17], Proposition 3.1, for initial data y 0 = 0 the explicit solution of (10) is given by y(t; s, y 0 ) 2 = e 2λt e 2λs y −2 0 + 2 t s e 2λτ β(τ )dτ , t > s.
Taking the limit s → −∞ we obtain two complete trajectories ±ξ, where that are unbounded when t tends to −∞ by Assumption (5). If ζ(t), t ∈ R, is a complete trajectory of (10) above of ξ(t), t ∈ R, the explicit solution formula implies It follows that which shows that solutions starting above of the complete trajectory ξ blow-up backwards in finite time and cannot be emanating from a bounded subset of R.
We observe that y(t) = 0, t ∈ R, is an equilibrium solution of (10), ξ(t) pullback attracts at time t all solutions emanating from initial data y 0 > 0 and −ξ(t) all solutions emanating from y 0 < 0. Moreover, the family of compact subsets is strictly invariant for the evolution process generated by (10). By the connectedness of its sections it follows that the global pullback attractor A ode of the ODE (10) is given by , t ∈ R. When restricted to the subspace of constant functions, the evolution process {U (t, s)| t ≥ s} generated by Problem (1) coincides with the evolution process generated by the ODE (10), which implies that and concludes the proof of the theorem.

4.2.
Non-autonomous damped wave equation. We consider the following initial value problem for the non-autonomous damped wave equation, u(x, t) = 0 x ∈ ∂Ω, t ≥ s, u(x, s) = u s (x) x ∈ Ω, s ∈ R, where s ∈ R and Ω ⊂ R n , n ∈ N, n ≥ 3, is a bounded domain with smooth boundary ∂Ω. We assume that the non-linearity f : R → R is continuously differentiable and satisfies |f (z)| ≤ c(1 + |z| p ), z ∈ R, lim sup for some constant c > 0 and 0 < p < 2 n−2 . Furthermore, the function β : R → R + is Hölder continuous and bounded from above and below by positive constants 0 < β 0 ≤ β 1 < ∞, We apply Theorem 2.2 to show that the evolution process generated by (11) possesses a pullback exponential attractor. Setting v := ∂ ∂t u and w := u v we rewrite Problem (11) in the abstract form ∂ ∂t w = A β (t)w + F (w) t > s, (15) w| t=s = w s w s ∈ V, s ∈ R, where the initial data w s = u s v s , and the phase space is V := H 1 0 (Ω) × L 2 (Ω). The norm in V is given by Furthermore, the operators are defined by A β (t) = A 1 + A 2 (t), The initial value problem (15) generates an evolution process {U (t, s)| t ≥ s} in the Banach space V , which is asymptotically compact and pullback strongly bounded dissipative. For details we refer to [4], Chapter 4 in [14], Section VI.4 in [6], [2] and [3].
We denote the evolution process generated by the linear homogeneous problem w| t=s = w s w s ∈ V, s ∈ R, by {C(t, s)| t ≥ s}. The following lemma was proved in [3] and yields the exponential decay of solutions of the linear homogeneous equation.
Lemma 4.5. Let {C(t, s)| t ≥ s} be the evolution process in V associated to the linear problem (16). Then, there exist constants C ≥ 0 and ω > 0 such that the norm of the operators is bounded by C(t, s) L(V ;V ) ≤ Ce −ω(t−s) ∀t ≥ s, t, s ∈ R.
The process {U (t, s)| t ≥ s} satisfies the integral equation U (t, s)w s = C(t, s)w s + t s C(t, τ )F (U (τ, s)w s )dτ = C(t, s)w s + S(t, s)w s (see [3] and [14]). Moreover, {U (t, s)| t ≥ s} is pullback strongly bounded dissipative and the pullback absorbing time corresponding to a bounded subset is independent of the time instant t ∈ R. For the proof of the following lemma we refer to [3].
Lemma 4.6. Let {U (t, s)| t ≥ s} be the evolution process in V generated by the initial value problem (15). Then, there exists a bounded uniformly pullback absorbing subset B ⊂ V , i.e., for every bounded set D ⊂ V there exists T D ≥ 0 such that To show that the family of operators {S(t, s)| t ≥ s} satisfies the smoothing property we establish several auxiliary results. We denote by X α , α ∈ R, the fractional power spaces associated to the operator A with domain D(A) = X 1 = H 1 0 (Ω)∩H 2 (Ω) in X := L 2 (Ω) (see [23] or [22]). Furthermore, let H s (Ω), s ∈ R + , be the fractional Sobolev spaces obtained by interpolation between the spaces H m (Ω) and L 2 (Ω), m ∈ N (see [1] or Section II.1.3 in [23]). Since the domain Ω is bounded we have the following continuous embeddings