Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces

In the paper, we gave a strengthening of our previous work in [32] (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B_{p,r}^s(\mathbb{R})$ with $s>\max \{1+1\mathrm{/}p,3\mathrm{/}2\}$ and $(p,r)\in [1,\mathrm{\infty }]\times [1,\mathrm{\infty })$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.