- Caisheng Chen
^{1}Email author, - Huaping Yao
^{1}and - Ling Shao
^{1}

**2010**:216760

https://doi.org/10.1155/2010/216760

© Caisheng Chen et al. 2010

**Received: **10 May 2010

**Accepted: **13 July 2010

**Published: **2 August 2010

## Abstract

## Keywords

## 1. Introduction

where and is a boundary domain in with smooth boundary . The assumptions on and will be made in the sequel.

More precisely, they obtained that the global existence of solution for (1.1)-(1.2) if one of the following assumptions was satisfied:

(ii) , the initial data is small.

Similar consideration can be found in [3–5]. In [6], Yang obtained the uniqueness of solution of the Laplacian wave equation (1.1)-(1.2) for . To the best of our knowledge, there are few information on the uniqueness of solution of (1.1)-(1.2) for and .

In this paper, we are interested in the global existence, the uniqueness, the continuity and the asymptotic behavior of solution for (1.1)-(1.2). The nonlinear term in (1.1) likes with and . Obviously, the sign condition fails to hold for this type of function.

For these purposes, we must establish the global existence of solution for (1.1)-(1.2). Several methods have been used to study the existence of solutions to nonlinear wave equation. Notable among them is the variational approach through the use of Faedo-Galerkin approximation combined with the method of compactness and monotonicity, see [7]. To prove the uniqueness, we need to derive the various estimates for assumed solution . For the decay property, like (1.5), we use the method recently introduced by Martinez [8] to study the decay rate of solution to the wave equation in , where is a bounded domain of .

This paper is organized as follows. In Section 2, some assumptions and the main results are stated. In Section 3, we use Faedo-Galerkin approximation together with a combination of the compactness and the monotonicity methods to prove the global existence of solution to problem (1.1)-(1.2). Further, we establish the uniqueness of solution by some a priori estimate to assumed solutions. The proof of asymptotic behavior of solution is given in Section 4.

## 2. Assumptions and Main Results

We first give some notations and definitions. Let be a bounded domain in with smooth boundary . We denote the space and for and and relevant norms by and , respectively. In general, denotes the norm of Banach space . We also denote by and the inner product of and the duality pairing between and , respectively. As usual, we write instead . Sometimes, let represent for and so on.

Let us state our assumptions on and .

with some and the nonnegative functions , , , where , .

A typical function is with the appropriate nonnegative functions and , where .

Definition 2.1 (see [7]).

Now we are in a position to state our results.

Theorem 2.2.

Further, if and , the solution satisfying (2.5)-(2.6) is unique.

Theorem 2.3.

The following theorem shows that the asymptotic estimate (2.9) can be also derived if assumption (2.8) fails to hold.

Theorem 2.4.

## 3. Proof of Theorem 2.2

In this section, we assume that all assumptions in Theorem 2.2 are satisfied. We first prove the global existence of a solution to problem (1.1)-(1.2) with the Faedo-Galerkin method as in [1, 2, 7, 9].

As it is well known, the system (3.3)-(3.4) has a local solution on some interval We claim that for any , such a solution can be extended to the whole interval by using the first a priori estimate below. We denote by the constant which is independent of and the initial data and .

Obviously, satisfies the estimates (2.5)-(2.6). Finally, using the standard monotonicity argument as done in [1, 7], we get that . This completes the proof of existence of solution .

Repeating the above procedure, we conduce that on and on . This ends the proof of uniqueness.

## 4. Proof of Theorem 2.3

Let us first state a well-known lemma that will be needed later.

Lemma 4.1 (see [10]).

### 4.1. The Proof of Theorem 2.3

This shows that is nonincreasing in .

where the fact that is nonincreasing is used.

This is (2.9) and we complete the proof of Theorem 2.3.

### 4.2. The Proof of Theorem 2.4

This ends the proof of Theorem 2.4.

## Declarations

### Acknowledgments

The authors express their sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. The work was supported by the Science Funds of Hohai University (Grant Nos. 2008430211 and 2008408306) and the Fundamental Research Funds for the Central Universities (Grant No. 2010B17914).

## Authors’ Affiliations

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