In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain of

Let

Here,

Denote by

Dissipative wave equations arise as an evolutionary mathematical model in various systems for the relevant physical application including electrodynamics, quantum mechanics, and nonlinear elasticity (see e.g., [

The case of supercubic growth rate is a bit more complicated since the uniqueness of energy weak solutions is unknown for the supercubic case (e.g., see [

Considering the strong solution, in 1983, Babin and Vishik [

As mentioned by many authors, the global attractor, however, does not provide an actual control of the convergence rate of trajectories and it can be sensitive to perturbations. In order to overcome these difficulties, Eden et al. introduced the notion of exponential attractor (see [

In the present paper, as a continuation of our previous works [

Our main result can be stated as follows.

Let

There exist a monotone increasing function

The above theorem will be proved in Section

Throughout the rest of the paper, denote by

Firstly, we recall the definitions about the weak solution and strong solution of problem (

(see [

For the well-posedness of strong solution, we have the following result.

Let

where

Therefore, the strong solution generates a

Finally, we state the existence of global attractor in strong energy space which will be used in Lemma

Let

There are two different approaches to find exponential attractors. The original one (cf. [

Let

The map

is Lipschitz continuous (with the metric inherited from

The map

where

for some

Then, the local dynamical system

In order to verify the local exponential attractor in

Let

We will verify all conditions of Lemma

Construct a bounded positively invariant set.

Decompose the solution

It is convenient to denote

Let the assumptions in Theorem

Moreover, the set

After vanishing the

On the other hand, by inequality (

Then, we define a positive invariant set by

It remains to show that

We will verify asymptotic regularity and Lipschitz continuity.

Let the assumptions in Theorem

We have estimates

where the constant

The mapping

(i) For any

From equation (

Taking inner product of (

Since

(ii) For

The first term is handled by estimate (

Therefore, applying Lemma

The data used to support the study are included within the article.

The authors declare that they have no conflicts of interest.

C. Liu was supported by the Natural Science Foundation of Jiangsu Province (Grant no. BK20170308) and NSFC (11801227); F. Meng was supported by NSFC (11701230); C. Zhang was supported by NSFC (11801228) and the dual creative (innovative and entrepreneurial) talents project in Jiangsu Province.