On the parabolic Harnack inequality for non-local diffusion equations

Abstract

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $d\ge \beta$, where $\beta \in (0,2]$ is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $t>0$ in dimensions $d\ge \beta$. This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $u_0 \in L^q_{loc}$ for q larger than the critical value $\tfrac{d}{\beta }$ of the elliptic operator $(-\Delta )^{\beta /2}$, a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $d=1$ provided $\beta >1$, since we prove that the local Harnack inequality holds if $d<\beta$.