UNIQUENESS OF GLOBALLY CONSERVATIVE WEAK SOLUTIONS FOR A WEAK DISSIPATION CAMASSA-HOLM TYPE EQUATION

In this paper, we prove the uniqueness of globally conservative solutions for Camassa-Holm type equation by the analysis of characteristics. Giving a conservative solution, we introduce a set of auxiliary variables tailored to this solution, and prove that these variables satisfy a suitable semilinear system of equations. By proving that this semilinear system has a unique solution, we eventually obtain the uniqueness of the conservative solution of the original equation in time weighted $H^{1}$ space.