Non-uniform continuity on initial data for the two-component b-family system in Besov space

Abstract: In this paper, we consider the Cauchy problem of a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system. It is shown that the solution map of the two-component b-family system is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max{1+\frac{1}{p}, \frac{3}{2}}$, $1\leq p, r< \infty$. Our result covers and extends the previous non-uniform continuity in Sobolev spaces $H^{s-1}(\mathbb{R})\times H^s(\mathbb{R})$ for $s>\frac{5}{2}$ (Nonlinear Anal., 2014) to Besov spaces.