Persistence property and the local well-posedness of the modified Camassa-Holm equation in critical Besov equation

In this paper, we first establish the local well-posednesss for the Cauchy problem of a modified Camassa-Holm (MOCH) equation in critical Besov spaces $B_{p,1}^{\frac{1}{p}}$ with $1\le p<+\mathrm{\infty }\mathrm{.}$ The obtained results improve considerably the recent result in \cite{Luo1}. Then we show the persiscence property of MOCH.