The Cauchy problem for fractional Camassa-Holm equation in Besov space

In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space $B_{2,1}^{s_0}$ with $s_0=2\nu -\frac{1}{2}$ for $\nu >\frac{3}{2}$ and $s_0=\frac{5}{2}$ for $1<\nu \le \frac{3}{2}$. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.