On the Cauchy problem for a weakly dissipative Camassa-Holm equation in critical Besov spaces

Abstract

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B_{p,r}^s$ with $s>1+\frac{1}{p}$ and $s=1+\frac{1}{p},r=1,p\in [1,\infty ).$ Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.

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