The Global Weak Solution for a Generalized Camassa-Holm Equation

Abstract

A nonlinear generalization of the famous Camassa-Holm model is investigated. Provided that initial value $u_0\in H^s(R)(1\le s\le 3/2)$ and $(1-{\partial }_x^2)u_0$ satisfies an associated sign condition, it is shown that there exists a unique global weak solution to the equation in space $u(t,x)\in L^2([0,+\infty ),H^s(R))$ in the sense of distribution, and $u_x\in L^{\infty }([0,+\infty )\times R)$.