The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

Abstract

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 $H^S\left(R\right)$ with $s>3/2$ is established via a limiting procedure. Provided that the initial value$u_0$ satisfies the sign condition and $u_0\in H^s\left(R\right)\left(s>3/2\right)$, it is shown that there exists a uniqueglobal solution for the equation in space $C\left([0,\infty \right);H^s\left(R\right))\cap C^1\left([0,\infty \right);H^{s-1}\left(R\right))$.