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

Abstract

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this 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 \(s>\max \{1+\frac{1}{p}, \frac{3}{2}\}\), \(1\le p, r< \infty \). Our result covers and extends the previous non-uniform continuity in Sobolev spaces \(H^{s-1}({\mathbb {R}})\times H^s({\mathbb {R}})\) for \(s>\frac{5}{2}\) to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces.