H\"older Continuity and Harnack estimate for non-homogeneous parabolic equations

TL;DR

This paper advances the understanding of non-homogeneous parabolic equations by establishing a forward-in-time Harnack inequality, leading to H"older continuity of solutions and a global Harnack estimate, building on prior work.

Contribution

It introduces a new forward-in-time intrinsic Harnack inequality for non-divergence form equations, enhancing previous results and providing alternative proofs for generalized inequalities.

Findings

Proves H"older continuity of solutions.

Establishes a global Harnack estimate.

Provides an alternative proof of generalized Harnack inequality.

Abstract

In this paper we continue the study on intrinsic Harnack inequality for non- homogeneous parabolic equations in non-divergence form initiated by the first author in [1]. We establish a forward-in-time intrinsic Harnack inequality, which in particular implies the H\"older continuity of the solutions. We also provide a Harnack type estimate on global scale which quantifies the strong minimum principle. In the time-independent setting, this together with [1] provides an alternative proof of the generalized Harnack inequality proven by the second author in [9].