  In this paper we obtain bounds for the decay rate for solutions to the
nonlocal problem $\partial_t u(t,x) = \int_{\R^n} J(x,y)[u(t,y) - u(t,x)] dy$.
Here we deal with bounded kernels $J$ but with polynomial tails, that is, we
assume a lower bound of the form $J(x,y) \geq c_1|x-y|^{-(n + 2\sigma)}$, for
$|x - y| > c_2$. Our estimates takes the form $\|u(t)\|_{L^q(\R^n)} \leq C
t^{-\frac{n}{2\sigma} (1 - \frac{1}{q})}$ for $t$ large.
