The H 1 ( R ) Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

Abstract

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space $C([0,\infty )$ × $\text{R})$ $\cap L^{\infty }([0,\infty )$; $H^1(\text{R}))$ provided that its initial value $u_0(x)$ belongs to the space $H^1(R)$. A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.