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In this paper, we prove that the existence and uniqueness of globally weaksolutions to the Cauchy problem for the weakly dissipative Camassa-Holmequation in time weighted $H^1$ space. First, we derive an equivalentsemi-linear system by introducing some new variables, and present the globallyconservative solutions of this equation in time weighted $H^1$ space. Second,we show that the peakon solutions are conservative weak solutions in $H^1.$Finally, given a conservative solution, we introduce a set of auxiliaryvariables tailored to this particular solution, and prove that these variablessatisfy a particular semilinear system having unique solutions. In turn, we getthe uniqueness of the conservative solution in the original variables.

Existence and uniqueness of the globally conservative solutions for a  weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$  space