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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 247841, 5 pages
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.
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.
Hakkaev and Kirchev  investigated the following generalized Camassa-Holm equation: where is an integer. When , (1) becomes the Camassa-Holm model (see ). 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 . 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 . This constitutes the objective of this work.
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 , Constantin and Molinet , Xin and Zhang , and Coclite et al. . It was shown in Constantin and Escher  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 . For other methods to investigate the problems involving various dynamic properties of the Camassa-Holm equation, the reader is referred to [13–16] 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.
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
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
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  (or see Coclite et al. ). 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).
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.
By the interpolation theorem, for all , we obtain where depends on and .
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
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