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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 615345, 27 pages
http://dx.doi.org/10.1155/2012/615345
Research Article

On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

Department of Mathematics, Southeast University, Nanjing 210018, China

Received 24 May 2012; Revised 17 July 2012; Accepted 17 July 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 Xiaopan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the blow-up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow-up of solutions can be obtained.

1. Introduction

In this paper, we investigate the following nonlinear wave equation: where is a bounded domain in with smooth boundary , is a Laplace operator, and indicates derivative of in outward normal direction of . In addition to, if and if , then . is a constant, are given in (A1) later.

In 1968, Greenberg et al. [1] first suggested and studied the following equation: Under the condition and higher smooth conditions on and initial data, they claimed the global existence of classical solutions for the initial boundary value problem of (1.2).

The multidimensional form of the following: was first studied by Clement [2, 3]. Exploiting the monotone operator method, he obtained the global existence of weak solutions for the initial boundary value problem of (1.2).

Our model comes from [4]. In [4], Yang has studied the global existence, asymptotic behavior, and blow-up of solutions for a nonlinear wave equations with dissipative term: with the same initial and boundary conditions as that of (1.1). In our model, we add damping and source terms which enhance the difficulty of proving the existence and decay of solution of (1.2).

More related studies of the damped hyperbolic equation with dissipative term or damping term can be found in papers [515].

The paper is organized as follows. In Section 2, we present some notations, and results needed later and main results. Section 3 contains the statement and the proofs of the decay of solutions. Section 4 gives the statement and the proofs of the blow-up of solutions.

2. Preliminaries

We first introduce the following abbreviations: Let () denote the -inner product. We denote the dual of by , with , and . Now we make the following assumptions:) and for some , if , else if , then , so that , , where is the optimal embedding constant.() The initial data (); If , then and if , then . () If , and if , then . Without loss of generality, we assume that .

And(), and for some , if , else if , then , so that , . where is the optimal embedding constant, and .() If , then and if , then .(), and if , then and if , then .

Throughout this paper, we use the embedding which implies when where is an optimal embedding constant. We introduce the following functionals: Because , we have

Theorem 2.1. Assume that ()–() hold, then problem (1.1) has a unique solution satisfying where .

Theorem 2.2. Assume that ()–() hold, is the local solution of the problem (1.1). And where , then is a global bounded solution, moreover, where is a constant.

Theorem 2.3. Assume that ()–() and hold, is the local solution of the problem (2.7). Consider (2.12) are satisfied, then where are constants.

Remark 2.4. When , we will use perturbed energy method, which is different to the method of the proof of Theorem 2.2, to prove Theorem 2.3.

Theorem 2.5. Assume that ()–(), () hold, and there exist some , then the solution of the problem (1.1) blows-up at the limited time .

3. Decay of Solutions

In this section, we prove Theorems 2.12.3. First, we give the following Lemma.

Lemma 3.1 (see [2]). Let be any bounded domain in , be an orthogonal basis in . Then for any , there exists a positive number such that for all .

Proof of Theorem 2.1. We look for approximate solutions of problem (1.1) of the form where is an orthogonal basis in , and also in , and the coefficients satisfy with Since is dense in and , we choose such that The above system of o.d.e. has a local solution defined in some interval . The following will prove that the can be substituted by some .
Multiply (3.3) by and summing up about , we get A simple integration of (3.5) over leads to where .
We now estimate the last two terms at the right-hand side of (3.6). Using Hӧlder inequality, Young's inequality, and the embedding theorem, we know there exist , satisfying that where Consider the following: assuming that .
Similarly, Using (3.6)–(3.10), we have Choosing in (3.11), we have where . Assuming , we have A simple integration of (3.13) over leads to this implies that Though may blow up, there exists satisfying where is independent of .
Moreover, By (3.17), By (3.16), From (3.16)–(3.19), we have So the solution of problem (3.3) exists on for each . On the other hand, we can extract a subsequence from , still denoted by , such that as . By (3.21), the Sobolev embedding theorem and the continuity of , for , as . By Lemma 3.1, (3.21)-(3.22), for any , there exist positive constant and independent of and , respectively, such that, as ,
By the arbitrariness of we get From the continuity of and (3.25) we know that a.e. on . With the same methods used above we easily get a.e. on . Integrating (3.3) over gets Exploiting (3.16)-(3.17), we have
Let in (3.26) and we deduce from (3.23), (3.27) and the Lebesgue-dominated convergence theorem that This implies is a local weak solution of problem (1.1). The proof of Theorem 2.1 is completed.

Secondly, we prove Theorem 2.2. First we give two lemmas. It is easy to prove what follows.

Lemma 3.2. The modified energy functional satisfies, along solutions of (1.1),

Lemma 3.3. Assume that ()–() hold, satisfying Then .

Proof. Since , then there exists (by continuity) such that , this gives By using (2.8), (2.9), (3.31), and Lemma 3.2, we easily have We then exploit (2.12), and (3.32) to obtain Using (3.33), we have where .
Therefore, for all . By repeating this procedure, and using the fact that the proof is completed.

Lemma 3.4. Assume that ()–() hold, (2.12) satisfy. Then the solution is global existence. More, exist positive constant has

Proof. It suffices to show that is bounded independently of . To achieve this, we use (2.9), (3.31), and Lemma 3.2 to get Since are positive. Therefore, Moreover, where is a positive constant, which depends only on .

Lemma 3.5. Assume that ()–() hold, (2.12) satisfy. Then exist has

Proof. Using Lemma 3.4, we have Using (3.34), we have So the proof is complete.

Lemma 3.6. Assume that ()–() hold, (2.12) satisfy. Then there exists a , having

Proof. By multiplying the differential equation in (1.1) by and integrating over , using integration by parts (2.9) and assumption (), we obtain So Using (), (3.42)-(3.43), we have Therefore,

Proof of Theorem 2.2. First, is also absolutely continuous, and we have A simple integration of (3.52) over leads to Using the upper inequality, (3.45), (3.46), and (3.52), we have Apply (3.42), (3.47), we can get (2.13).
Using (3.31) and Lemma 3.3, we get . (2.13) implies . So when , we have and . It is that (2.14) is satisfied. Theorem 2.2 is complete.
Following we will prove Theorem 2.3. For this purpose we set where is a positive constant and

Lemma 3.7. Let be small enough. Then there exist two positive constants and such that

Proof. By Lemma 3.4 and Young's inequality, a direct computation gives Similarly, we have provided that is small enough.

Lemma 3.8. Assume that the conditions of Theorem 2.3 hold, then the function satisfies, along the solution of (1.1),

Proof. Applying equations of (1.1), we see Exploiting the assumption () and Young's inequality, we have Exploiting (3.63) and (3.62), we get Choosing satisfies and satisfies at the above; the proof of (3.61) is completed.

Proof of Theorem 2.3. Using (3.61) and Lemma 3.7, we have Choosing satisfies . So we have A simple integration of (3.68) over leads to Exploiting Lemma 3.7 again, we have where are constants. The proof of Theorem 2.3 is complete.

4. Blow-Up of Solutions

Proof of Theorem 2.5. Assuming that the solution of (1.1) is global, we have So we set where are constants and will be given later.
Consider Choosing satisfies the following condition in (4.4): so we have Moreover, we have Define where will be given later, and Multiplying (1.1) by and a direct computation yield