  Popowicz system, as the interacting system of Camassa-Holm and
Degasperis-Procesi equations, has attracted some attention in recent years. In
this paper, we first study the local well-posedness for the cauchy problem of
Popowicz system in nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with
$s> \max\{2, \frac{1}{p}+\frac{3}{2}\}$ or $(s=2, 2\leq p \leq \infty, 1\leq
r\leq 2)$. Moreover, a new blow-up criterion and global existence with
different initial values are obtained.
