Backward Harnack inequality and Hamilton estimates for heat type equations

TL;DR

This paper establishes a backward-in-time Harnack inequality for positive solutions of heat equations on compact manifolds, enabling comparison of solution values at different space-time points in both temporal directions.

Contribution

It introduces a novel backward Harnack inequality based on Hamilton's gradient estimates, extending the classical inequality without additional restrictions.

Findings

Backward Harnack inequality for heat equations on compact manifolds

Comparison of solution values at different space-time points in both directions

Potential applications in analysis of heat equations

Abstract

Based on gradient estimates for the heat equation by Hamilton, we discover a backward in time Harnack inequality for positive solutions on compact manifolds without further restrictions such as boundedness or vanishing boundary value for solutions. Contrary to the usual Harnack inequality, it allows comparison of the values of a solution at two different space time points, in both directions of time. In view of the importance of the usual Harnack inequality, further application is expected.