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 s−1 p,r (R)×B s p,r (R) with s > max{1+ 1 p , 3 2 }, 1 ≤ p, r < ∞. Our result covers and extends the previous non-uniform continuity in Sobolev spaces H s−1 (R) × H s (R) for s > 5 2 (Nonlinear Anal., 2014) to Besov spaces.