Well-posedness and blow-up phenomena for an integrable three-component Camassa–Holm system

Abstract

This paper studies the Cauchy problem for an integrable three-component Camassa–Holm system. We first establish the local well-posedness with initial condition in Besov spaces. Then we prove a blow-up criteria by arguing inductively with respect to the regularity index. Moreover, we derive a Riccati-type differential inequality by using the structure of equations, and also prove a new blow-up criteria with sufficient conditions on initial condition, whose proof is based on the conservative property of potential densities along the characteristic.