Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa–Holm equation in time weighted H1(R) space

Abstract

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa–Holm equation in time weighted H1 space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted H1 space. Second, given a globally conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system has a unique solution. In turn, we get the uniqueness of the conservative solutions in the original variables. Finally, we show that the peakon solutions are conservative weak solutions in H1.