The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

Abstract

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that $(1-{\partial }_x^2)u_0\in M^{+}\left(R\right)$, $u_0\in H^1(R),$ and $u_0\in L^1\left(R\right)$, the existence and uniqueness of the global weak solution to the equation are shown to be true.