On a three-Component Camassa-Holm equation with peakons

Abstract

In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case(with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we showthat its solutions are analytic in both variables, globally in space andlocally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extensionresults for functions on the half-line and several symmetry preserving properties ofthe equations under discussion.