An Intrinsic Harnack inequality for some non-homogeneous parabolic equations in non-divergence form

TL;DR

This paper proves a scale-invariant Harnack inequality for certain non-homogeneous parabolic equations, extending previous elliptic results with a novel geometric covering approach tailored to the equations' nonlinearity.

Contribution

It introduces a new intrinsic geometry and a modified covering argument to establish a Harnack inequality for a class of inhomogeneous parabolic equations, extending prior elliptic results.

Findings

Established a scale-invariant Harnack inequality for specific parabolic equations.

Developed a new geometric covering technique suited for nonlinear inhomogeneous equations.

Extended elliptic Harnack inequalities to a parabolic setting with nonlinearity.

Abstract

In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the stack of cubes covering argument which is tailored for the nonlinearity that we consider.