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

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 + 1/p and s = 1 + 1/p, r = 1, p epsilon [1,infinity). 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.