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The local and global existence of solutions for a generalized Camassa-Holm equation. (English) Zbl 1237.35139

Summary: A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev space \(H^s(\mathbb{R})\) with \(s > 3/2\) is established via a limiting procedure. Provided that the initial value \(u_0\) satisfies the sign condition and \(u_0 \in H^s (\mathbb{R})\) (\(s > 3/2\)), it is shown that there exists a unique global solution for the equation in space \(C([0, \infty); H^s(\mathbb{R})) \cap C^1([0, \infty); H^{s-1}(\mathbb{R}))\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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