Harnack's Inequality for Parabolic Nonlocal Equations

TL;DR

This paper establishes a nonlocal version of Harnack's inequality for parabolic equations, including weak Harnack inequality and local boundedness, without requiring solutions to be globally positive.

Contribution

It introduces a novel nonlocal Harnack inequality for parabolic equations, extending classical results to nonlocal operators without positivity constraints.

Findings

Proves a nonlocal Harnack inequality for parabolic equations

Establishes weak Harnack inequality for solutions

Demonstrates local boundedness of solutions

Abstract

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.