On the parabolic Harnack inequality for non-local diffusion equations

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 = beta, where beta. (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 = 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 u0. L q loc for q larger than the critical value d beta of the elliptic operator (- 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.