Nonuniform dependence for the two-component Camassa–Holm-type system with higher-order nonlinearity in Besov spaces

Abstract

Considered herein is the Cauchy problem of the two-component Camassa–Holm-type system with higher-order nonlinearity. Based on the local well-posedness results of this problem, we establish that the solution map $z_0\mapsto z(t)$ of this problem in the periodic case is not uniformly continuous in Besov spaces $B_{p,r}^s(\mathbb{𝕋})\times B_{p,r}^{s-1}(\mathbb{𝕋})$ with and $B_{2,1}^{3∕2}(\mathbb{𝕋})\times B_{2,1}^{1∕2}(\mathbb{𝕋})$ through the method of approximate solutions. This phenomenon reveals that the local well-posedness results of the solutions to this problem in the corresponding Besov spaces cannot be obtained by a sole contraction principle argument.