About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 247841, 5 pages
http://dx.doi.org/10.1155/2013/247841
Research Article

Several Dynamic Properties of Solutions to a Generalized Camassa-Holm Equation

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 31 March 2013; Accepted 12 May 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Zheng Yin. 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

For a nonlinear generalization of the Camassa-Holm equation, we investigate the dynamic properties of solutions for the equation under the assumption that the initial value lies in the space . A one-sided upper bound estimate on the first-order spatial derivative, bound estimate, and a space-time higher-norm estimate for the solutions are obtained.

1. Introduction

Hakkaev and Kirchev [1] investigated the following generalized Camassa-Holm equation: where is an integer. When , (1) becomes the Camassa-Holm model (see [2]). The local well-posedness in the Sobolev space with is established, and sufficient conditions for the stability and instability of the solitary wave solutions are given in [1]. However, the estimate of strong solutions and one-sided upper bound estimate on the first-order spatial derivative for the solutions are not discussed in [1]. This constitutes the objective of this work.

Like the Camassa-Holm equation (see [1, 2]), (1) has the conservation law which plays an important role in our further investigations.

In fact, many scholars have paid their attentions to the study of the Camassa-Holm equation. The existence of global weak solutions is established in Constantin and Escher [3], Constantin and Molinet [4], Xin and Zhang [5], and Coclite et al. [6]. It was shown in Constantin and Escher [7] that the blowup occurs in the form of breaking waves. Namely, the solution remains bounded, but its slope becomes unbounded in finite time. After wave breaking, the solution is continued uniquely either as a global conservative weak solution [8, 9] or as a global dissipative solution [10, 11]. Exact traveling wave solutions for the Camassa-Holm equation are presented in [12]. For other methods to investigate the problems involving various dynamic properties of the Camassa-Holm equation, the reader is referred to [1316] and the references therein.

In this paper, we investigate several dynamic properties of strong solutions for the generalized Camassa-Holm equation (1) in the case where is an odd natural number and the assumption . The results obtained in this work include a one-sided upper bound estimate on the first-order derivatives of the solution, a space-time higher-norm estimate, and the bound estimate.

The rest of this paper is organized as follows. Section 2 states the main result. Several lemmas are given in Section 3 where the proof of main result is completed.

2. Main Result

Let be a nonnegative integer and . In this case, the Cauchy problem for (1) is written in the form which is equivalent to where the operator and

We introduce a result presented in [1] for problem (3).

Lemma 1 (see [1]). Suppose that with constant . Then there is a real number such that the problem (3) has a unique solution satisfying

Now we state the main result of this paper.

Theorem 2. Let with . Then the solution of problem (3) has the following properties.(a)There exists a positive constant depending on and such that the one-sided norm estimate on the first-order spatial derivative holds (b)Let , , and . Then there exists a positive constant depending only on , and such that the following estimate holds: (c)There exists a constant depending only on such that

3. Proof of Main Result

From the conservation law (2), we have Differentiating the first equation of problem (4) with respect to and writing , we obtain

Lemma 3. Let , and . Then there exists a positive constant depending only on , and , such that the space higher integrability estimate holds where is the unique solution of problem (3).

Proof. The proof is a variant of the proof presented in Xin and Zhang [5] (or see Coclite et al. [6]). Let be a cut-off function such that and Letting , and yields from which we get
Multiplying (11) by , using the chain rule, and integrating over , we have From (16), we get Using the Hölder inequality, (2), and (15) yields Using (10), we obtain Applying (10), the Hölder inequality, and , we have where is a constant depending on and .
From (20) and (21), we deduce that there exists a positive constant depending on and , such that from which we get From inequalities (18)–(19) and (23), we obtain (12).

Lemma 4. There exists a positive constant depending only on and such that If , it holds that

Proof. We have The first inequality of (24) is proved in Lemma 3 (see (22)). For the second inequality in (24), we have which together with (22) results in (24).
In fact, we have From (29)–(31), we obtain (25).
Using completes the proof of (26).

From Lemma 1, we know that for any with , there exist a maximal and a unique strong solution to problem (3) such that

Consider the differential equation

Lemma 5. Assume , and let be the maximal existence time of the solution to problem (3). Then there exists a unique solution to problem (32). In addition, the map is an increasing diffeomorphism of with for .

Proof. Using Lemma 1, we obtain and . Therefore, we know that functions and are bounded, Lipschitz in space, and in time. The existence and uniqueness theorem for differential equations guarantees that problem (32) has a unique solution .
From (32), we get Furthermore, For every , the Sobolev imbedding theorem gives rise to Therefore, there exists a constant such that for . The proof is completed.

Using (11) and (33), we get It follows from (36) that Using Lemma 5 and (37), we have

Using the first equation of problem (4) and Lemma 4, for an arbitrary integer , we have from which we get

By the interpolation theorem, for all , we obtain where depends on and .

From Lemma 3 and (38) and (41), we complete the proof of Theorem 2.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).

References

  1. S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 30, no. 4–6, pp. 761–781, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Constantin and J. Escher, “Global weak solutions for a shallow water equation,” Indiana University Mathematics Journal, vol. 47, no. 4, pp. 1527–1545, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Constantin and L. Molinet, “Global weak solutions for a shallow water equation,” Communications in Mathematical Physics, vol. 211, no. 1, pp. 45–61, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411–1433, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. M. Coclite, H. Holden, and K. H. Karlsen, “Global weak solutions to a generalized hyperelastic-rod wave equation,” SIAM Journal on Mathematical Analysis, vol. 37, no. 4, pp. 1044–1069, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a Lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10–12, pp. 1511–1549, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. O. G. Mustafa, “Global dissipative solutions of the generalized Camassa-Holm equation,” Journal of Nonlinear Mathematical Physics, vol. 15, no. 1, pp. 96–115, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  11. H. Holden and X. Raynaud, “Global conservative solutions of the generalized hyperelastic-rod wave equation,” Journal of Differential Equations, vol. 233, no. 2, pp. 448–484, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H.-H. Dai, “Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods,” Wave Motion, vol. 28, no. 4, pp. 367–381, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Lai and Y. Wu, “Global solutions and blow-up phenomena to a shallow water equation,” Journal of Differential Equations, vol. 249, no. 3, pp. 693–706, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Lai and A. Wang, “The well-posedness of solutions for a generalized shallow water wave equation,” Abstract and Applied Analysis, vol. 2012, Article ID 872187, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Y. Lai, “The global weak solution for a generalized Camassa-Holm equation,” Abstract and Applied Analysis, vol. 2013, Article ID 838302, 6 pages, 2013. View at Publisher · View at Google Scholar