A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity

Abstract

Considered herein is the Cauchy problemfor a modified Camassa-Holm equation with cubic nonlinearity. The local well-posedness in Besov space $B^s_{2,1}$ with the critical index $s=5/2$ is established. Then a lower bound for the maximal time of existence of its solutions is found. Withanalytic initial data, the solutions to this Cauchy problem are analytic in both variables, globally in space and locally in time, which extends the result of Himonas and Misiołek [A. Himonas, G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575---584] to more general $\mu$-version equations and systems.