The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.

We consider a stochastic viscoelastic wave equation with nonlinear damping and source terms

For the deterministic case on viscoelastic wave equation, many authors studied the following problem:

In fact, lots of investigators have paid attention to the viscoelastic wave equation, which has its origin in the mathematical description of viscoelastic materials. The dynamic properties of viscoelastic materials are of great importance as they appear in many applications to natural sciences. The general viscoelastic wave equation has the following form:

Under the consideration of random environment, some authors investigated the following stochastic wave equation with nonlinear damping and source terms:

Recently, Wei and Jiang [

As we know, no one considers the stochastic viscoelastic wave equation (

In contrast with the model in [

This paper is organized as follows. In the next section, we recall some preliminaries related to assumptions and definitions for the solutions of the stochastic equations. In Section

Let

For all

One assumes that

In this paper,

To simplify the computations, we assume that the covariance operator

Assume that

In this section, we establish the local existence and uniqueness of solution to problem (

Denote

For any

Let

Let

Fix

For notational convenience, we omit

Assume (

Let

Since

Due to integration by parts, we have

From (

Let

From (

Next, we will prove the uniqueness of the solution. If there is another solution

Finally, we state that

Moreover, we still fix

Assume that (

Assume that

In this section we prove our main result for

For each

In order to prove our blow-up result, we rewrite (

First we give a lemma.

Assume that (

Using

Let

Let

If

Assume that (

there exists a positive time

For the lifespan

Define

Using (

Inserting (

In view of (

Hence, substituting (

Note that

Next, we can choose

Therefore, (

We choose

Therefore we have

Let

The proof of Theorem

(1) In the deterministic case of

(2) Our results have included the case which is without viscoelastic term (i.e.,

In this section we show that solution of (

Assume that (

For any

Since

Either

Or

Consequently we have

Taking the expectation of (

On the other hand, we have

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by NSF of China 11272277 and 11226188 and FRF for the Central Universities of China 2013ZZGH027.