  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 $B^{s}_{p,r}$ 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.
