Journal of Mathematics

Volume 2013 (2013), Article ID 578094, 4 pages

http://dx.doi.org/10.1155/2013/578094

Research Article

On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation

Wenbin Zhang,¹ Jiangbo Zhou,² Lixin Tian,² and Sunil Kumar³

¹Taizhou Institute of Science and Technology, NUST, Taizhou, Jiangsu 225300, China

²Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

³Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India

Received 10 January 2013; Accepted 21 January 2013

Academic Editor: Ji Gao

Copyright © 2013 Wenbin Zhang et al. 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.

It is shown that if is a sufficiently smooth solution to a two-dimensional nonlinear wave equation such that there exists with supp , for , then .

1. Introduction

In this paper, we consider the following two-dimensional nonlinear wave equation:

where , , , are arbitrary positive constants. Equation (1) was recently derived by Gottwald [1] for large scale motion from the barotropic quasigeostrophic equation as a two-dimensional model for Rossby waves. He [2] showed that (1) has traveling wave solutions via the homotopy perturbation method. Using a subequation method, the traveling wave solutions are also studied by Fu et al. [3]. Aslan [4] constructed solitary wave solutions and periodic wave solutions to (1) by the Exp-function method.

For and in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma. Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ([6–11]). Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically. Recently, Bustamante et al. [13] showed that sufficiently smooth solutions of the ZK equation that have compact support for two different times are identically zero.

The purpose of this paper is to investigate the support of solutions to (1). To solve the problem, we mainly use the ideas of [12–15]. The main result is as follows.

Theorem 1. Assume that and , if is a solution of (1) such that

then, .

2. Preliminary Estimates

Lemma 2 (see [13]). Assume that and . (i) If , then

(ii) if , then

where , , and .

Lemma 3. Assume that , if is a solution of (1) such that ; then, is bounded in .

Proof. Assume that is a decreasing function with if and if . Let for and . It is easy to check that and

Multiplying (1) by and integrating by parts in , we obtain

Applying Gronwall Lemma and the Monotone Convergence Theorem, we have

This proves that is bounded in .

Applying Lemma 2 with and , we have that is bounded in . Here, we used the fact that . This completes the proof of the lemma.

Lemma 4. Assume that , , and is a solution of (1). (i) If , then is bounded in ; (ii) if , then is bounded in .

Proof. Letting and a solution to (1), we have

Multiplying (8) by and integrating by parts in , we obtain

Note that and

It follows from (9) that

Since and is bounded in , applying Lemma 2 with and , we have that is also bounded in .

Similarly, we can prove that is bounded in . Let ; then, is a solution to (1) and satisfies , and therefore is bounded in . This proves (i).

Now, we prove (ii). Let ; then, is also a solution of (1) and satisfies the hypothesis of (i). This proves (ii) and completes the proof of the lemma.

Remark 5. In particular, if the conditions for and given in (i) and (ii), respectively, are satisfied, then is bounded in .

Lemma 6 (see [13]). Let , is a function such that is bounded in , and . Then, for all and all , the functions and are absolutely continuous in with derivatives and a.e. , respectively.

Lemma 7. Assume that , , and , if is a function such that is bounded in and . Then,

Proof. Let and ; then,

Taking the spatial Fourier transform in (14) and applying Lemma 6, we have

where

According to (15), when , we have

and when , we choose to write

Therefore, we have

Appling Plancherel formula, we have inequality (12).

Similarly, letting , we can also have (13). This completes the proof of the lemma.

Lemma 8 (see [12, 13]). Assume that , and , if is a solution to (1) such that

then, .

Proof. The proof is similar to that of Theorem  1.1 in [12], and we omit the details.

3. Proof of the Main Result

Assume that for where is a nondecreasing function such that for and for . Let ; then, . According to Lemma 7, we obtain that

where and

Note that the derivatives of are supported in the interval ; then,

where is dependent on and .

Combining (21) with (23), we obtain

Applying Lemma 4 with , we have that

then

Since , taking such that , we have

Note that for ; we have

Letting , we obtain

and this proves that in .

Next, we will prove that in . Let ; then,

where .

In fact, Let ; it is easy to check that also satisfies the hypotheses of this theorem, and then we find that in

Thus, there exists such that for all . Applying Lemma 8, we complete the proof of Theorem 1.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11171135), the Natural Science Foundation of Jiangsu (no. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), as well as the Taizhou Social Development Project (no. 2011213).

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