Growth of Solutions with Positive Initial Energy to Systems of Nonlinear Wave Equations with Damping and Source Terms

Abstract

We consider initial-boundary conditions for coupled nonlinear wave equations with damping and source terms. We prove that the solutions of the problem are unbounded when the initial data are large enough in some sense.

1. Introduction

In this work, we consider the following initial-boundary value problem: where is a bounded domain with smooth boundary in , ; ; are given functions to be specified later.

Throughout this paper, we define by where , are nonnegative constants and .

This type of problems not only is important from the theoretical point of view, but also arises in material science and physics that deal with system of nonlinear wave equations.

Ye [1] obtained the local existence and the blowup of the solution of problem (1), for . In the absence of the strong damping terms, problem (1) becomes Wu et al. [2] obtained the global existence and blowup of the solution of problem (3) under some suitable conditions. Fei and Hongjun [3] considered problem (3) and improved the blowup result obtained in [2], for a large class of initial data in positive initial energy, using the same techniques as in Payne and Sattinger [4] and some estimates used firstly by Vitillaro [5]. Recently, Pişkin and Polat [6, 7] studied the local and global existence, energy decay, and blowup of the solution of problem (3). Also, for more information about (1) and (3), see [2, 3, 7].

The many problems associated with (1) are studied from various aspects in many papers [8–13].

In this work, we will consider the blowup property in infinity time, that is, exponential growth.

This work is organized as follows. In Section 2, we state the local existence result. In Section 3, we establish that the energy will grow up as an exponential as time goes to infinity, provided that the initial data are large enough or , where and are defined in (9) and (15).

2. Preliminaries

In this section, we introduce some notations and lemmas and local existence theorem needed in the proof of our main results. Let and denote the usual norm and norm, respectively.

Concerning the functions and , we take where are constants and satisfies According to the above equalities they can easily verify that where

We have the following result.

Lemma 1 (see [14]). There exist two positive constants and such that is satisfied.

We define the energy function as follows: where , .

Lemma 2 (see [7]). is a nonincreasing function for and

Lemma 3 (Sobolev-Poincare inequality [15]). Let be a number with or , then there is a constant such that

Next, we state the local existence theorem [1, 7].

Theorem 4 (local existence). Suppose that (5) holds. Then there exist , satisfying and further , . Then problem (1) has a unique local solution

3. Exponential Growth

In this section, we will prove that the energy is unbounded when the initial data are large enough in some sense. Our techniques of proof follow very carefully the techniques used in [16].

Lemma 5 (see [3]). Suppose that (5) holds. Then there exists such that for any the inequality holds.

For the sake of simplicity and to prove our result, we take and introduce where is the optimal constant in (14). Next, we will state a lemma which is similar to the one introduced firstly by Vitillaro in [5] to study a class of a single wave equation.

Lemma 6 (see [3]). Suppose that (5) holds. Let be the solution of problem (1). Assume further that and Then there exists a constant such that for all .

Theorem 7. Suppose that (5) and hold. Then any solution of problem (1) with initial data satisfying grows exponentially.

Proof. We set From (10) and (20) we get hence we have .
We consider the following functional: for small to be specified later.
Our goal is to show that satisfies a differential inequality of the form This, of course, will lead to exponential growth.
By taking a derivative of (22) and using (1), it follows that From (9) and (20), it follows that Inserting (25) into (24), we get Then using (18), we obtain where . It is clear that , since . In order to estimate the last two terms in (27), we use the following Young inequality: where , , such that . Consequently, applying the above inequality we have Inserting estimates (29) into (27), we have where .
Since , from the embedding and embedding , we have for some positive constants and . Using the algebraic inequality and since , we get where . Similarly Inserting (33) and (34) into (30), we have Now, once and are fixed, we can choose small enough such that Consequently (35) takes the form where  .
Then we have On the other hand, applying Hölder inequality, we obtain Young inequality gives Since , algebraic inequality (32) yields Note that Combining with (37) and (42), we arrive at Integrating differential inequality (43) between and gives the following estimate for : The proof of Theorem 7 is completed.

Remark 8. When , by setting , the similar result is obtained by applying the same arguments in the proof of Theorem 7.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.