About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 121489, 6 pages
http://dx.doi.org/10.1155/2014/121489
Research Article

On the Stability to a Generalized Degasperis-Procesi Equation

1Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
2Department of Mathematics, Sichuan Normal University, Chengdu 610066, China

Received 28 August 2013; Accepted 11 December 2013; Published 6 February 2014

Academic Editor: Yonghong Wu

Copyright © 2014 Haibo Yan 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

A nonlinear generalized Degasperis-Procesi equation is investigated. Assuming that the strong solution of the equation is bounded in the sense of -norm and the initial data belong to the space , we prove that the solutions are stable in the space .

1. Introduction

Coclite and Karlsen [1] investigated the following generalized Degasperis-Procesi equation: When and satisfies or where is a positive constant, the existence and stability of entropy weak solutions belonging to the class are established for (1) in paper [1].

The objective of this paper is to study the generalized Degasperis-Procesi equation where is a positive constant, is a polynomial of order , and . When and , (4) reduces to the classical Degasperis-Procesi model [210]. Assuming that there exists a strong solution to (4), which is bounded in its existence time interval , and the initial value of (4) lies in , we will prove that the strong solutions of the equation are stable in the space (see Theorem 8 in Section 3). From the authors’ knowledge, this is a new result for (4).

This paper is organized as follows. Section 2 gives several lemmas. The main result and its proof are presented in Section 3.

2. Several Lemmas

We consider the Cauchy problem of (4) in the following form: Applying the operator to the first equation of problem (5), we obtain where . Letting , we get

Lemma 1. The solution of problem (5) with satisfies where and . Moreover, there exist two constants and depending only on such that

Proof. Letting and and using (4), we obtain and Using the Parseval identity and (10), we obtain (8) and (9).

Remark 2. When , from (8), we cannot obtain inequality (9).

Lemma 3. If and , it holds that where is a constant independent of and .

Proof. Using the assumption and Lemma 1, we have . Using (7), we get Since the function is a polynomial of order and , combining Lemma 1 derives that (11) holds.

Lemma 4. Assume that and are two solutions of (4) with initial data , respectively. Then, for any , it holds that where depends on , , , , , and .

Proof. We have which completes the proof.

We define as a function which is infinitely differentiable on such that , for , and . For any number , we let . Then we know that is a function in and Assume that the function is locally integrable on . We define an approximation function of as We get as almost everywhere.

We state the concept of a characteristic cone. For any , we define . Let represent the cone . We let represent the cross section of the cone by the plane , . Set , where , , and for an arbitrary . The space of all infinitely differentiable functions with compact support in is denoted by .

Lemma 5 (see [11]). Let the function be bounded and measurable in cylinder . If and , then the function satisfies .

Lemma 6 (see [11]). Let be bounded. Then the function satisfies the Lipschitz condition in and , respectively.

Using the methods presented in [11], we have the following result.

Lemma 7. If is a strong solution of problem (6), , and , it holds that where is an arbitrary constant.

Proof. Let be a twice differential function on the line . We multiply the first equation of problem (6) by the function , where . Integrating over and transferring the derivatives with respect to and to the test function , for any constant , we obtain in which we have used .
Integration by parts yields that Let be an approximation of the function and set . Using the properties of the , from (20) and (21), and sending , we have which completes the proof.

3. Main Result

Generally speaking, we cannot get the boundedness of strong solutions for problem (6). This is why we assume that the strong solutions of problem (6) possess boundedness in order to establish the stability for the problem. Now we state our main result as follows.

Theorem 8. Assume that there exist strong solutions and for problem (5) or (6). Let be the maximum existence time for the solutions. If , , and the initial data , it holds that where depends on ,, , , and the coefficients of polynomial .

Proof. For , we assume that outside the cylinder Let where and . The function is defined in (15). Note that
Following Kruzkov’s device of doubling the variables presented in [11], from Lemma 7, and choosing , we have Similarly, it has It follows from (27) and (28) that
We will prove the following inequality: We observe that the first two terms of inequality (29) can be represented in the form From Lemma 6, we know that satisfies the Lipschitz condition in and , respectively. By the choice of , we have outside the region Considering the estimate and the expression of function , we have where the constant does not depend on . Using Lemma 5, we obtain as . The integral does not depend on . In fact, substituting , , , and and noting that we have Hence Since we obtain Using Lemma 5, we have as . Using (35), we have From (33),(37), (39), and (40), we prove that inequality (30) holds.
Set We define and choose two numbers and . In (30), we choose where When is sufficiently small, we note that function outside the cone and outside the set . For , we have the relations
Applying (41)–(45) and (30), we have the inequality Using Lemma 4 and letting and , we obtain
By the properties of the function for , we have where is independent of . Letting we get from which we obtain Similarly, we have It follows from (51) and (52) that
Sending and and using from (47), (48), and (53)-(54), we have Applying the Gronwall inequality yields the desired result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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

References

  1. G. M. Coclite and K. H. Karlsen, “On the well-posedness of the Degasperis-Procesi equation,” Journal of Functional Analysis, vol. 233, no. 1, pp. 60–91, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta, Eds., vol. 1, no. 1, pp. 23–37, World Scientific, Singapore, 1999.
  3. D. Henry, “Infinite propagation speed for the Degasperis-Procesi equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 755–759, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Z. Lin and Y. Liu, “Stability of peakons for the Degasperis-Procesi equation,” Communications on Pure and Applied Mathematics, vol. 62, no. 1, pp. 125–146, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Lundmark and J. Szmigielski, “Multi-peakon solutions of the Degasperis-Procesi equation,” Inverse Problems, vol. 19, no. 6, pp. 1241–1245, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Liu and Z. Yin, “Global existence and blow-up phenomena for the Degasperis-Procesi equation,” Communications in Mathematical Physics, vol. 267, no. 3, pp. 801–820, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Lai and Y. Wu, “A model containing both the Camassa-Holm and Degasperis-Procesi equations,” Journal of Mathematical Analysis and Applications, vol. 374, no. 2, pp. 458–469, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Lenells, “Traveling wave solutions of the Degasperis-Procesi equation,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 72–82, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. O. G. Mustafa, “A note on the Degasperis-Procesi equation,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 10–14, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. Yin, “Global weak solutions for a new periodic integrable equation with peakon solutions,” Journal of Functional Analysis, vol. 212, no. 1, pp. 182–194, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. N. Kruzkov, “First order quasi-linear equations in several independent variables,” Mathematics of the USSR-Sbornik, vol. 10, no. 2, pp. 217–243, 1970.