Advances in Mathematical Physics
Volume 2015 (2015), Article ID 523254, 5 pages
http://dx.doi.org/10.1155/2015/523254

Research Article

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

Erhan Pişkin

Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

Received 27 November 2014; Accepted 15 February 2015

Academic Editor: Ivan Avramidi

Copyright © 2015 Erhan Pişkin. 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.

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.

References

Y. Ye, “Existence and decay estimate of global solutions to systems of nonlinear wave equations with damping and source terms,” Abstract and Applied Analysis, vol. 2013, Article ID 903625, 9 pages, 2013.

J. Wu, S. Li, and S. Chai, “Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 3969–3975, 2010.

L. Fei and G. Hongjun, “Global nonexistence of positive initial-energy solutions for coupled nonlinear wave equations with damping and source terms,” Abstract and Applied Analysis, vol. 2011, Article ID 760209, 14 pages, 2011.

L. E. Payne and D. H. Sattinger, “Saddle points and instability of nonlinear hyperbolic equations,” Israel Journal of Mathematics, vol. 22, no. 3-4, pp. 273–303, 1975.

E. Vitillaro, “Global nonexistence theorems for a class of evolution equations with dissipation,” Archive for Rational Mechanics and Analysis, vol. 149, no. 2, pp. 155–182, 1999.

E. Pişkin, “Blow up for coupled nonlinear wave equations with weak damping terms,” International Journal of Differential Equations and Applications, vol. 12, no. 4, pp. 131–137, 2013.

E. Pişkin and N. Polat, “Global existence, decay and blow up solutions for coupled nonlinear wave equations with damping and source terms,” Turkish Journal of Mathematics, vol. 37, no. 4, pp. 633–651, 2013.

W. Liu, “Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions,” Applied Mathematics Letters, vol. 38, pp. 155–161, 2014.

W. Liu and S. Li, “General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions,” Zeitschrift für angewandte Mathematik und Physik, vol. 65, no. 1, pp. 125–134, 2014.

Q. Bie and C. Luo, “Blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source terms,” Mathematica Applicata, vol. 24, no. 3, pp. 479–487, 2011.

Y. Xiong, “Blow-up and polynomial decay of solutions for a viscoelastic equation with a nonlinear source,” Zeitschrift für Analysis und ihre Anwendungen, vol. 31, no. 3, pp. 251–266, 2012.

C. Wenying and X. Yangping, “Blow-up and general decay of solutions for a nonlinear viscoelastic equation,” Electronic Journal of Differential Equations, no. 12, 11 pages, 2013.

S.-T. Wu, “General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions,” Zeitschrift für angewandte Mathematik und Physik, vol. 63, no. 1, pp. 65–106, 2012.

S. A. Messaoudi and B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 277–287, 2010.

R. A. Adams and J. J. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics, Academic Press, 2nd edition, 2003.

B. Said-Houari, “Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” Zeitschrift für Angewandte Mathematik und Physik, vol. 62, no. 1, pp. 115–133, 2011.