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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<s3 2 is established under the assumptions u 0 H s and ||u 0x || L <. The local well-posedness of solutions for the equation in the Sobolev space H s (R) with s>3 2 is also developed.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35D30Weak solutions of PDE
35A01Existence problems for PDE: global existence, local existence, non-existence
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
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