On the Harnack inequality for time-fractional and more general non-local in time subdiffusion equations

TL;DR

This paper proves the Harnack inequality for positive solutions of one-dimensional nonlocal in time subdiffusion equations, including time-fractional diffusion, filling a gap in the understanding of these equations.

Contribution

It establishes the Harnack inequality for a broad class of nonlocal in time subdiffusion equations in one dimension, where it was previously unknown.

Findings

Harnack inequality holds for these equations in one space dimension

Classical Harnack inequality does not hold in higher dimensions for these equations

Extends understanding of solution behavior for nonlocal in time diffusion models

Abstract

In this paper we establish the Harnack inequality for globally positive local solutions to a general class of nonlocal in time subdiffusion equations in one space dimension, which includes time-fractional diffusion equations with time order less than one. It is already known that for these equations the classical Harnack inequality does not hold if the space dimension is greater than or equal to two. Here, we complete the analysis, by providing a positive result in one space dimension.