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In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a nonlinear viscoelastic wave equation with strong damping, linear damping and source terms. Then we study the global attractors of the equation.

We know that viscoelastic materials have memory effects. These properties are due to the mechanical response influenced by the history of the materials. As these materials have a wide application in the natural science, their dynamics are of great importance and interest. The memory effects can be modeled by a partial differential equation. In recent years, the behaviors of solutions for the PDE system have been studied extensively, and many achievements have been obtained. Many authors have focused on the problem of existence, decay and blow-up for the last two decades, see [

In [

and they proved the global existence, uniqueness and exponential stability of solutions and existence of the global attractor.

In [

where

is a non-autonomous term, called a symbol, and ρ is a real number such that

In this work, we intend to study the following initial-boundary problem:

where

near damping term

A direct computation yields

Thus, the original memory term can be written as

and we get a new system

with the initial conditions

and the boundary conditions

The rest of this paper is organized as follows. In Section 2, we first obtain the priori estimates, then in Section 3, we prove the existence of the global attractors.

For convenience, we denote the norm and scalar product in

In this section, we present some materials needed in the proof of our results, state a global existence result, and prove our main result. For this reason, we assume that

(G1)

(G2)

(G3) There exists a constant

Lemma 1. Assume (G1), (G2) and (G3) hold, let

and

here

Proof. We multiply

By using Holder inequality, Young’s inequality and Poincare inequality, we get

and

and

For the first term on the right side (2.5), by using (G1), (G2) and (G3), we have

where

For the second term on the right side (2.5), by using Holder inequality and Young’s inequality, we get

So, we have

By using Poincare inequality, we obtain

and

and

By using Holder inequality and Young’s inequality, we obtain

Then, we have

That is

Next, we take proper

Taking

where

From

Then

So, there exists

Lemma 2. Assume (G1), (G2) and (G3) hold, let

and

Here

Proof. We multiply

By using Holder inequality, Young’s inequality and Poincare inequality, we get

and

and

For the first term on the right side (2.23), by using (G1), (G2) and (G3), we have

where

For the second term on the right side (2.23), by using Holder inequality and Young’s inequality, we get

so, we have

By using Poincare inequality, we have

and

And using Interpolation Theorem, we have

By using Holder inequality and Young’s inequality, we have

Then, we have

That is

Next, we take proper

Taking

where

From

then

So, there exists

Theorem 1. Assume (G1), (G2) and (G3) hold, let

and

Proof. By the method of Galerkin and Lemma 1 and Lemma 2, we can easily obtain the existence of solutions. Next, we prove the uniqueness of solutions in detail.

Assume that

where

By multiplying the equation by

here

and

by using (G1), (G2) and (G3), we have

By using Poincare inequality, we have

and

By using Holder inequality, Young’s inequality and Poincare inequality, we have

then, we have

That is

Taking

By using Gronwall inequality, we have

So we get

Theorem 2. Let X be a Banach space, and

1)

2) It exists a bounded absorbing set

3) When

Therefore, the semigroup operators

Theorem 3. Under the assume of Theorem 1, equations have global attractor

where

1)

2)

Proof. Under the conditions of Theorem 1, it exists the solution semigroup

1) From Lemma 1 to Lemma 2, we can get that

the ball

This shows that

2) Furthermore, for any

So, we get

3) Since

So, the semigroup operators

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.