Uniform Exponential Attractors for Non-Autonomous Strongly Damped Wave Equations

In this paper, we study the existence of exponential attractors for strongly damped wave equations with a time-dependent driving force. To this end, the uniform Hölder continuity is established to the variation of the process in the phase apace. In a certain parameter region, the exponential attractor is a uniformly exponentially attracting time-dependent set in the phase apace, and is finite-dimensional no matter how complex the dependence of the external forces on time is. On this basis, we also obtain the existence of the infinite-dimensional uniform exponential attractor for the system.


Introduction
In this paper, we study the following non-autonomous strongly damped wave equation on a bounded domain ( ) ( ) .
we make the following assumptions on functions ( ) ( ) , with 0 2; where 1 2 , c c are positive constants.And we assume that the external force g belongs to the space ( ) ( ) for some given (possibly large) constant M .Wave equations, describing a great variety of wave phenomena, occur in the extensive applications of mathematical physics.Equation (1.1) can be regarded as a perturbed equation of a continuous Josephson junction where ( ) sin , see [1].There is a large literature on the asymptotic behavior of solutions for strongly damped wave equations (see, for instance, [1]- [9]).In [9], the author showed the uniform boundedness of the global attractor for large strongly damping and obtained an estimate of the upper bound of the Hausdorff dimension of an attractor for strongly damped wave Equation (1.1) when g is independent of t .But when the equations depend explicitly on t , the case can be complex.
Recently, motivated by [6], the authors have given a new explicit algorithm allowing to construct the exponential attractor, and this method makes it possible to consider more general processes in applications [10] [11].
An exponential attractor  is a compact semi-invariant set of the phase space whose fractal dimension is finite and which attracts exponentially the images of the bounded subsets of the phase space Φ .In non-autonomous dynamical systems, instead of a semigroup, we have a so-called (dynamical) process ( ) , U t τ depending on two parameters , t τ ∈  (or , t τ ∈  for discrete times).The asymptotic behavior of non-autonomous dy- namical systems is essentially less understood and, to the best of our knowledge, the finite-dimensionality of the limit dynamics was established only for some special (e.g.quasiperiodic) dependence of the external forces on time.Indeed, there exists, at the present time, one of the different approaches for extending the concept of a global attractor to the non-autonomous case which is based on the embedding of the non-autonomous dynamical system into a larger autonomous one by using the skew-product flow.This approach naturally leads to the socalled uniform attractor un  which remains time-independent in spite of the fact that the dynamical system now depends explicitly on the time, see [12].We note that however the uniform attractor reduces to an autonomous system via the skew-product flow.It seems natural to generalize the concept of an exponential attractor to the non-autonomous case, see [11] [13] [14].But in all these articles, the uniform attractor's approach was used in order to construct an exponential attractor for the non-autonomous system considered and, consequently, an (uniform) exponential attractor remained time-independent.Since, under this approach, an exponential attractor should contain the uniform attractor, all the drawbacks of uniform attractors (artificial infinite-dimensionality and high dynamical complexity) described above are preserved for exponential attractors.
In the present article, we study exponential attractors of the system (1.1) based on the concept of a non-autonomous (pullback) attractor.Thus, in the approach, an exponential attractor is also time-dependent.To be more precise, a family ( ) The fractal dimension of all the sets ( ) is finite and uniformly bounded with respect to : t 2) There exist a positive constant β and a monotonic function Q such that, for every We emphasize that the convergence in (1.6) is uniform with respect to t ∈  and, consequently, under this approach, we indeed overcome the main drawback of global attractors [13].
This article is organized as follows.In Section 2, we first provide some basic settings and show the absorbing and continuous properties in proper function space about Equation (1.1).In Section 3 and Section 4, we prove the existence of the uniform attractor and exponential attractor of Equation (1.1), respectively.Finally, we prove the existence of infinite-dimensional exponential attractor, and compare it with the non-autonomous exponential attractor in Section 5.

Preliminaries
We will use the following notations as that in Pata and Squassina [15].Let A be the (strictly) positive operator on ( ) and the compact, dense injections , .
Define a new weighted inner product and norm in E as where µ is chosen as ) Obviously, the norm 3) is equivalent to the usual norm ( ) ( ) where k is chosen as ( ) and then the system (1.1) can be written as where ; we only need to prove lemma 2.1 for any 2 By (2.4) and (2.6), elementary computation shows ( ) The proof is completed. Lemma 2.2 Let assumptions (1.2)-(1.5)be satisfied.For any initial data E τ ϕ ∈ , there exists a positive con- stant ρ depending only on the coefficients of (1.3) and (2.4) and Ω such that the following dissipative esti- mate holds: where Q is a monotonic function and where the positive number 1 M depends also on M (but is indepen- dent of the concrete choice of g ).
∈ be the solution of the system (2.5) with the initial value ( ) ( ) Taking the inner product ( ) By (1.2), (1.3) and Poincaré inequality, there exist two positive constants 1 2 , 0 By (2.4) and (2.6), This completes the proof. Theorem 2.1 Given any 0 b > and for the solutions of (2.5) with any two initial data 1 2 , , , e , , for some ( ) The proof is similar to Theorem 2 in [15]. Theorem 2.2 For the solutions of (2.5) with different external forces 1 g and 2 g satisfying (1.5) and with the initial data 1τ ϕ and 2 E τ ϕ ∈ , the following contiuity holds: , , e d , , where C and 2 K are independent of t M , and .

τ
The proof is similar to Lemma 4 in [5].

Existence of the Uniform Attractor
The dissipativity property obtained in Lemma 2.2 yields the existence of an absorbing set for the process ( ) , g U t τ on E .In the following section, we assume that 6 c σ ≥ holds, where 6 c is specified in (3.11).
Theorem 3.1 The process and we introduce the splitting ( ) ( ) ( ) ( ) where ( ) , and ( ) We now define the families of maps

t f u Aq A f u A p kp A f u A p A f u u A p k A f u A p t
for the first term on the right-hand side of (3.9), we have ) By (3.9), (3.10), and Lemma 2.2, there is 1 ( ) , By (1.2), (1.3) and (2.8), (2.9), from (3.12), we obtain , , , , , , , .
Lemma 2.2 and (3.13) imply that ( ) , w ϕ ρ = , we will prove that there exists 0 , w ρ , we thus obtain ( ) ( ) due to Gronwall and Poincaré inequalities, then Since the embedding E  is compact, (3.13), (3.14) and the following lemma imply that Lemma 3.1 (see [16]) Let X be a complete metric space and Λ be a subset in X , such that , the same arguments in the Equation (3.4) lead to where k R is an isomorphism of : ≥ also possesses a uniform attractor .

Existence of Exponential Attractors
The main result of this section is the following theorem.Theorem 4.1 Let the function f and the external force g satisfy the above assumptions.Then, for every external force g enjoying (1.5), there exists an exponential attractor ( ) of the dynamical process (1.1) which satisfies the following properties: 1) The sets ( ) U t τ and translation-invariant with respect to time-shifts: where , , , T h∈  is the group of temporal shifts, ( )( ) ( ). hg T t g t h = + 2) They satisfy a uniform exponential attraction property as follows: there exist a positive constant 2 β and a monotonic function Q (both depending only on M ) such that, for every bounded subset B of E , we have ( . 3) The sets ( ) where the constant 1 C is independent of t and g .4) The map ( ) is Hölder continuous in the following sense: ( , e d , where the positive constants 2 3 , C β and η are independent of 1 2 , g g and t , , , where 3 C and 1 η are also independent of t g, and s .Proof.Firstly, we construct a family of exponential attractors for the discrete dynamical processes associated with Equation (2.5).According to Lemma 2.2, it only remains to construct the required exponential attractors for initial data belonging to the where 0 R is a sufficiently large number depending only on M given in (1.5), is a uniform absorbing set for all the processes ( ) , g U t τ generated by Equation (1.1).Moreover, from Theorem 2.1, Theorem 2.
, , , where the constant C depends on M , and is independent of , 0.
Moreover, for every 0, T > we also have ( where q′ is a positive number and the positive constant T C′ depends on T but is independent of τ , t and s . Proof.Note that there is a 0 0 s > such that  U t τ and translation-invariance with respect to time-shifts is similar to [11] [13].Estimate (4.2) follows in a standard way from Lemma 2.2, Theorem 3.1 for the processes ( ) , g U t τ .Thus, it only remains to verify the finiteness of the fractal dimension of ( ) In order to prove this, we first note that, according to the Hölder continuities Theorem 2.1 in [13] and (4.7), we have ε > , and some constant C and C′ which are independent of t .The proof of Theorem 4.1 is completed.

Infinite-Dimensional (Uniform) Exponential Attractor and Non-Autonomous Exponential Attractor
Finally, we compare the non-autonomous exponential attractor ( ) obtained above with the so-called infinite-dimensional (uniform) exponential attractor constructed in [11] [13].To the existence of the uniform attractor for strongly damped wave equations, we use the results in [4] and [5] as a model example. Let : , Using the standard product flow in [4] and [5], for every external forces g satisfying (1.5), we can embed the dynamical process ( ) , g U t τ into the autonomous dynamical system ( ) acting on the extended phase space ( )  It is also known that the uniform attractor ( ) un g  exists under the relatively weak assumption that the hull ( ) , but, unfortunately, for more or less general external forces , g its Hausdorff and fractal dimensions are infinite.Instead, the following estimate for its Kolmogorov's ε -entropy holds, see [4].

2 L
 with the standard inner products and norms, respec- tively,

,
Ap Aq and integrate over Ω to obtain

(
component of the Cartesian product is called the uniform attractor associated with problem (1.1).

Proposition 5 . 1
Let the above assumptions hold and the hull ( ) g  of the initial external forces be compact.Then, Equation (1.1) possesses the uniform attractor ( ) un g  and its ε -entropy can be estimated in terms of the ε -entropy of the hull 2 and Theo- rem 3.1, it follows Lipschitz continuity and smooth properties for the difference of two , , Let the assumptions of Theorem 4.1 hold and let, in addition, the hull