Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations

TL;DR

This paper proves a Harnack inequality for nonnegative solutions to complex doubly nonlinear parabolic equations that involve both local and nonlocal operators, using advanced comparison and positivity expansion techniques.

Contribution

It introduces a novel approach combining comparison principles and positivity expansion to establish Harnack inequalities for mixed local and nonlocal parabolic equations.

Findings

Established Harnack inequality for the class of equations

Developed new comparison and positivity expansion methods

Extended classical results to mixed local and nonlocal operators

Abstract

In this paper, we establish the Harnack inequality of nonnegative weak solutions to the doubly nonlinear mixed local and nonlocal parabolic equations. This result is obtained by combining a related comparison principle, a local boundedness estimate, and an integral Harnack-type inequality. Our proof is based on the expansion of positivity together with a comparison argument.