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

Abstract

For Besov spaces $B^s_{p,r}(\mathbb R)$ with $s > \max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\mathbb R)$ to $C([0,T]; B^s_{p,r}(\mathbb R))$. The proof of non-uniform dependence is based on approximate solutions, the Littlewood-Paley decomposition and estimates on the linear transport equation.