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

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) (R) x B-p,r(s) (R) with s > max{1 + 1/p, 3/2}, 1 <= p, r <= infinity. Our result covers and extends the previous non-uniform continuity in Sobolev spaces Hs-1 (R) x H-s (R) for s > 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.