Local well-posedness and persistence properties for a model containing both Camassa-Holm and Novikov equation. (English) Zbl 1304.35220
Summary: This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities,
\[
u_t - u_{xxt} + ku_x + au^m u_x = (n + 2)u^n u_x u_{xx} + u^{n+1} u_{xxx},
\]
where \(k, a \in \mathbb R\) and \(m, n \in \mathbb Z^+\). This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of
strong solutions for this equation in Sobolev space \(H^s (\mathbb R)\) with \(s > \frac{3}{2}\) is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space \(H^s (\mathbb R)\) with \(1 <s\leq \frac{3}{2}\) is established.
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