- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Mathematics
Volume 2013 (2013), Article ID 578094, 4 pages
On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation
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.
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 .
In this paper, we consider the following two-dimensional nonlinear wave equation: where , , , are arbitrary positive constants. Equation (1) was recently derived by Gottwald  for large scale motion from the barotropic quasigeostrophic equation as a two-dimensional model for Rossby waves. He  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. . Aslan  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 , 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  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.  showed that sufficiently smooth solutions of the ZK equation that have compact support for two different times are identically zero.
Theorem 1. Assume that and , if is a solution of (1) such that then, .
2. Preliminary Estimates
Lemma 2 (see ). 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 ). 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
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.
Proof. The proof is similar to that of Theorem 1.1 in , 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 .
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.
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).
- G. A. Gottwald, “The Zakharov-Kuznetsov equation as a twodimensional model for nonlinear Rossby wave,” http://arxiv.org/abs/nlin/0312009.
- J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.
- Z. Fu, S. Liu, and S. Liu, “Multiple structures of two-dimensional nonlinear Rossby wave,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 383–390, 2005.
- A. A. Aslan, “Generalized solitary and periodic wave solutions to a -dimensional Zakharov-Kuznetsov equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1421–1429, 2010.
- V. E. Zakharov and E. A. Kuznetsov, “On three-dimensional solitons,” Journal of Experimental and Theoretical Physics, vol. 39, pp. 285–286, 1974.
- B. K. Shivamoggi, “The Painlevé analysis of the Zakharov-Kuznetsov equation,” Physica Scripta, vol. 42, no. 6, pp. 641–642, 1990.
- F. Linares and A. Pastor, “Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1060–1085, 2011.
- A. V. Faminskiĭ, “The Cauchy problem for the Zakharov-Kuznetsov equation,” Differential Equations, vol. 31, no. 6, pp. 1002–1012, 1995.
- H. A. Biagioni and F. Linares, “Well-posedness results for the modified Zakharov-Kuznetsov equation,” in Nonlinear Equations: Methods, Models and Applications, vol. 54 of Progress in Nonlinear Differential Equations and Their Applications, pp. 181–189, Birkhäuser, Basel, Switzerland, 2003.
- F. Linares and A. Pastor, “Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,” SIAM Journal on Mathematical Analysis, vol. 41, no. 4, pp. 1323–1339, 2009.
- F. Linares, A. Pastor, and J.-C. Saut, “Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton,” Communications in Partial Differential Equations, vol. 35, no. 9, pp. 1674–1689, 2010.
- M. Panthee, “A note on the unique continuation property for Zakharov-Kuznetsov equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 59, no. 3, pp. 425–438, 2004.
- E. Bustamante, P. Isaza, and J. Mejía, “On the support of solutions to the Zakharov-Kuznetsov equation,” Journal of Differential Equations, vol. 251, no. 10, pp. 2728–2736, 2011.
- J. Bourgain, “On the compactness of the support of solutions of dispersive equations,” International Mathematics Research Notices, no. 9, pp. 437–447, 1997.
- C. E. Kenig, G. Ponce, and L. Vega, “On the support of solutions to the generalized KdV equation,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 19, no. 2, pp. 191–208, 2002.