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A model containing both the Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1202.35231
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:
35Q53 KdV equations (Korteweg-de Vries equations)
35D30 Weak solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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