The continuous dependence and non-uniform dependence of the rotation Camassa-Holm equation in Besov spaces

In this paper, we first establish the local well-posedness and continuous dependence for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces $B_{p,r}^s$ with $s>1+1\mathrm{/}p$ or $s=1+1\mathrm{/}p,p\in [1,+\mathrm{\infty }),r=1$ by a new way: the compactness argument and Lagrangian coordinate transformation, which removes the index constraint $s>3\mathrm{/}2$ and improves our previous work \cite{guoy1}. Then, we prove the solution is not uniformly continuous dependence on the initial data in both supercritical and critical Besov spaces.