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

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 $s>\max \{1+\frac{1}{p},\frac{3}{2}\}$, $1\le p,r<\mathrm{\infty }$. 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.