On the parabolic Harnack inequality for non-local diffusion equations

TL;DR

This paper investigates the Harnack inequality for non-local in time diffusion equations, revealing that memory effects and dimensionality critically influence the inequality's validity, especially in fractional time derivatives.

Contribution

It constructs a counter-example showing failure of the Harnack inequality in certain dimensions and establishes conditions under which a non-local Harnack inequality holds.

Findings

Counter-example for dimensions d ≥ β where Harnack fails

Harnack inequality holds if initial data is sufficiently integrable

Critical dimension phenomenon affects diffusion behavior in higher dimensions

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 $d<\beta$.