Fractional decay bounds for nonlocal zero order heat equations

In this paper we obtain bounds for the decay rate for solu- tions to the nonlocal problem ∂t u(t, x) = Rn 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) ≥ c1 |x − y|^−(n+2σ) , for |x − y| > c2 . Our estimates takes the form u(t) Lq (Rn ) ≤ C t^{− n/2σ (1−1/q)} for t large