Nonlocal heat equations: decay estimates and Nash inequalities

We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of L\'evy type. We obtain these estimates in terms only of the behaviour of the kernel at infinity, without any information of its behaviour at the origin. This includes bounded and unbounded transition probability densities. An equivalence between the decay and a restricted Nash inequality is shown. We also prove that $\lim_{t\to \infty}\|u(t)\|_\infty=0$. Finally we deal with nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}\varphi(u)=0$.