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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 H?lder 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.

In this paper, we study the following non-autonomous strongly damped wave equation on a bounded domain

where

where

for some given (possibly large) constant

Wave equations, describing a great variety of wave phenomena, occur in the extensive applications of mathe- matical physics. Equation (1.1) can be regarded as a perturbed equation of a continuous Josephson junction where

Recently, motivated by [

An exponential attractor

In the present article, we study exponential attractors of the system (1.1) based on the concept of a non-au- tonomous (pullback) attractor. Thus, in the approach, an exponential attractor is also time-dependent. To be more precise, a family

1) The fractal dimension of all the sets

2) There exist a positive constant

We emphasize that the convergence in (1.6) is uniform with respect to

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.

We will use the following notations as that in Pata and Squassina [

with domain

and consider the family of Hilbert spaces

Then we have

and the compact, dense injections

In particular, naming

We recall the continuous embedding

and the interpolation results: given

and let

Equation (1.1) is equivalent to the following initial value problem in the Hilbert space E

where

It is well known (see, e.g., [

Define a new weighted inner product and norm in

for any

where

Obviously, the norm

Let

where

and then the system (1.1) can be written as

where

Lemma 2.1 For any

where

Proof. Since

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

where

Proof. Write

value

By (1.2), (1.3) and Poincaré inequality, there exist two positive constants

By (2.4) and (2.6),

where

By Gronwall’s inequality, we have an absorbing property:

This completes the proof. ,

Theorem 2.1 Given any

for some

The proof is similar to Theorem 2 in [

Theorem 2.2 For the solutions of (2.5) with different external forces

where

The proof is similar to Lemma 4 in [

The dissipativity property obtained in Lemma 2.2 yields the existence of an absorbing set for the process

Theorem 3.1 The process

Proof. We consider

and we introduce the splitting

and

We now define the families of maps

First step: We prove that

where

Similar to Lemma 2.1, we have

where

Similar to Lemma 2.2, applying (3.7) and Young, Poincaré, Gronwall inequalities, we obtain

Now we multiply (3.5) by

with

Then from (3.8) we have

i.e.,

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

let

By (1.2), (1.3) and (2.8), (2.9), from (3.12), we obtain

Lemma 2.2 and (3.13) imply that

Second step: Let

Multiply (3.3) by

due to Gronwall and Poincaré inequalities, then

Since the embedding

Lemma 3.1 (see [

with

Third step: Let

Then from (3.15), Lemma 2.2, and the compactness of

It is easy to see that the process

defined by (2.5) has the following relation with

where

Since the process

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 ex- ternal force g enjoying (1.5), there exists an exponential attractor

1) The sets

where

2) They satisfy a uniform exponential attraction property as follows: there exist a positive constant

3) The sets

where the constant

4) The map

where the positive constants

where

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 ball

where

Lemma 4.1 Let the above assumptions on Equation (1.1) hold. Then, for every

where the constant

where

Proof. Note that there is a

and

where all the constants depend on

Now, we can define the exponential attractors for continuous time by the following formula

with respect to

for all

for a given

Finally, we compare the non-autonomous exponential attractor

Let

where

Using the standard skew product flow in [

It is known that

It is also known that the uniform attractor

Proposition 5.1 Let the above assumptions hold and the hull

for some positive constants

Definition 5.1 [

1) Entropy estimate:

2) Semi-invariance: for every

3) Uniform exponential attraction property: there exists a positive constant

[

Theorem 5.1 Let the assumptions of Theorem 4.1 hold and let, in addition, the hull

Remark 1 When

Equation (1.1) reduces to the damped wave equation modeling the Josephson junction in superconduction which was studied by many authors (see [

Remark 2 When

if we assume that the function

This work is supported by the National Natural Science Foundation of China (11101265) and Shanghai Edu- cation Research and Innovation Key Project of China (14ZZ157).