Fractional decay bounds for nonlocal zero order heat equations

Abstract

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