The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-HolmEquation

Abstract

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space $C\left(\right[0,T),H^s\left(R\right))\cap C^1\left(\right[0,T),H^{s-1}\left(R\right))$ with . In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space $H^s\left(R\right)$ with $1\le s\le 3/2$ is developed.