Abstract: In this paper, the asymptotic behavior of the weak solution (u_t)_t\ge0 to the non-local Cauchy problems as stated in (1) is considered. Only using lower bounds of jumping kernel J(x,y) for large |x-y|, it is obtained that \|u_t\|_p\le c(t)\|u_0\|_q with any 1\le q<p<\infty and large t. Explicit and sharp formulas for c(t) are also given.