Intrinsic Harnack's inequality for a general nonlinear parabolic equation in non-divergence form

TL;DR

This paper establishes an intrinsic Harnack's inequality for a broad class of nonlinear parabolic equations, extending classical results to more general forms with stable constants and optimal exponents.

Contribution

It introduces a unified approach to prove Harnack's inequality for nonlinear parabolic equations in non-divergence form, generalizing previous specific cases.

Findings

Proved Harnack's inequality for generalized parabolic equations

Achieved results for the optimal range of exponents

Ensured stability of the constants involved

Abstract

We prove the intrinsic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.