Local well-posedness and blow-up result for weakly dissipative Camassa-Holm equations

Abstract

In this paper, we consider the Cauchy problem of the weakly dissipative Camassa-Holm equation. We prove the local well-posedness in the critical inhomogeneous Besov space $B^{3/2}_{2,1}( \mathbb R )$. This result depends on the apriori estimate of the nonlinear transport equation. Moreover, we show result for the finite time blowing up solution of the the weakly dissipative Camass-Holm equation which unifies previously known result. The proof relies on geometrical approach for the weakly dissipative Camass-Holm equation.