Weak Harnack inequality for a mixed local and nonlocal parabolic equation

TL;DR

This paper establishes a weak Harnack inequality with a tail term for supersolutions of a mixed local and nonlocal parabolic equation, using purely analytic methods including energy estimates and Moser iteration.

Contribution

It introduces a novel analytic approach for mixed local and nonlocal parabolic equations, adapting Bombieri's lemma and proving key inequalities.

Findings

Proves a weak Harnack inequality with tail term

Develops reverse H"older inequality for supersolutions

Uses Moser iteration and Bombieri's lemma in a new context

Abstract

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse H\"older inequality and a logarithmic estimate for weak supersolutions.