The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

Abstract

A nonlinear partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev space $H^s\left(R\right)$ with $s>3/2$. Although the $H^1$-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space $H^s$ with $1\le s\le 3/2$ is proved under the assumptions $u_0\in H^s$ and .