Continuity of the data-to-solution map for the FORQ equation in Besov Spaces

Abstract

For Besov spaces $B^s_{p,r}(\rr)$ with $s>\max \{2+\frac{1}{p},\frac{5}{2}\}$, $p\in (1,\infty ]$ and $r\in [1,\infty )$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\rr)$ to $C([0,T]; B^s_{p,r}(\rr))$. The proof of non-uniform dependence is based on approximate solutions and the Littlewood-Paley decomposition.