The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa–Holm system

Abstract

This paper studies the Cauchy problem for a two-component high-order Camassa–Holm system proposed in Escher and Lyons (2015). First, we investigate the local well-posedness of the system in the Besov spaces Bp,rs×Bp,rs−2 with s>max{3+1p,72,4−1p} and p,r∈[1,∞]. Second, by means of the Littlewood–Paley decomposition technique and the conservative property at hand, we derive a blow-up criteria for the strong solution. Finally, we study the Gevrey regularity and analyticity of the solutions to the system in the Gevrey–Sobolev spaces. In particular, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping.