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Zbl 1202.35231
Lai, Shaoyong; Wu, Yonghong
A model containing both the Camassa-Holm and Degasperis-Procesi equations. (English)
[J] J. Math. Anal. Appl. 374, No. 2, 458-469 (2011). ISSN 0022-247X

Summary: A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the $H^{1}$-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space $H^s$ with $1<s \leqslant \frac 32$ is established under the assumptions $u_{0} \in H^s$ and $|| u_{0x}||_{L^{\infty }}< \infty $. The local well-posedness of solutions for the equation in the Sobolev space $H^s(R)$ with $s > \frac 32$ is also developed.

MSC 2000:: *35Q53 KdV-like equations
35D30
35A01
35A02

Keywords: Camassa-Holm equation; Degasperis-Procesi; local well-posedness; weak solution

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Zentralblatt MATH Berlin [Germany]