^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if
*p* >
*m*, and a global result for
*p* ≤
*m*.

A purely elastic material has a capacity to store mechanical energy with no dissipation (of the energy). A complete opposite to an elastic material is a purely viscous material. The important thing about viscous materials is that when the force is removed it does not return to its original shape. Materials which are outside the scope of these two theories will be those for which some, but not all, of the work done to deform them can be recovered. Such materials possess a capacity of storage and dissipation of mechanical energy. This is the case for viscoelastic material. The dynamic properties of viscoelastic materials are of great importance and interest as they appear in many applications to natural sciences. Many authors have given attention to this problem for quite a long time, especially in the last two decades, and have made a lot of progress.

In [

where

For the problem (1.1) in

where

energy such that

In the absence of the viscoelastic term

In [

in a bounded domain and

In [

where

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

where

for the problem (1.6), the memory term

and we consider the strong damping term

Now, we shall add a new variable

A direct computation yields

Thus, the original memory term can be written as

and we get a new system

with the initial conditions

and boundary conditions

The paper is organized as follows. In Section 2, we first prove the blow up result, and then in Section 3, we prove the global existence result.

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 local existence result, which can be established, combining the argument of [

(G1)

(G2)

(G3) There exists a constant

We start with a local existence theorem which can be established by the Faedo-Galerkin methods. The interested readers are referred to Cavalcanti, Domingos Cavalcanti and Soriano [

Theorem 2.1. Assume (G1) holds. Let

Then for any initial data

with compact support, problem (1.10) has a unique solution

for some

Lemma 2.2. Assume (G1), (G2), (G3) and (2.1) hold. Let

where

Proof. By multiplying the Equation in (1.10) by

For the fourth term on the left side (2.4), by using (1.11), (G2) and (G3), we have

where

Then, we obtain

So, we have

where

Our main result reads as follows.

Lemma 2.3. Suppose that (2.1) holds. Then there exists a positive constant

for any

Proof. If

So, we obtain

If

Therefore (2.9) follows.

We get

and use, throughout this paper, C to denote a generic positive constant.

As a result of (2.3) and (2.5), we have

Corollary 2.4. Suppose that (2.1) holds. Then, we have

for any

Lemma 2.5. (

where

Proof. We set

By taking a derivative of

If

Then, we have

If

The proof is completed.

Next, we have the following theorem concerning blow up.

Theorem 2.6. Assume (G1), (G2), (G3) and (2.1) hold. Let

if

Proof. From (2.2), we have

consequently, we have

Similar to [

where

By multiplying (1.10) by

By using Holder inequality and Young’s inequality to estimate the fourth term on the right hand side of (2.19)

for some number

Then, we have

that is

By using Holder inequality and Young’s inequality to estimate the last two terms on right hand side of (2.24), we obtain

and

and

Substituting (2.24), (2.25) and (2.26) and to (2.23), we have

by taking

by taking proper

so, we have

From (2.16), we have

Then, hence (2.31) yields

where

Writing

where

From (2.3) and (G1) we have

writing

at this point, we choose

By using Holder inequality and Young’s inequality, we next estimate

and

which implies

where

By using

According to (2.36) and (2.41), we get

where

So, we know

In this section, we show that solution of (1.10) is global if

Lemma 3.1. For

Proof.

so,

Theorem 3.2. Assume (G1), (G2) and (G3) hold. Let

for any

Proof. Similar to [

from (2.3), we have

By differentiating

By using Holder inequality and Young’s inequality, we next estimate

Setting

Substituting (3.5) to (3.3), we have

so, there exists a small enough constant

Then, by using Gronwall inequality and continuation principle, we complete the proof of the global existence result.

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