  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
$B^s_{p,r}$ with $s>1+1/p$ or $s=1+1/p,\ p\in[1,+\infty),\ r=1$ by a new way:
the compactness argument and Lagrangian coordinate transformation, which
removes the index constraint $s>3/2$ 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.
