Differential Harnack Estimates for Time-dependent Heat Equations with Potentials

TL;DR

This paper establishes differential Harnack inequalities and gradient estimates for positive solutions of time-dependent heat equations with potentials, advancing understanding of their behavior over time.

Contribution

It introduces new differential Harnack inequalities and gradient estimates specifically for heat equations with potentials, which were not previously available.

Findings

Proved a differential Harnack inequality for positive solutions.

Established a gradient estimate for solutions.

Enhanced understanding of heat equation behavior with potentials.

Abstract

In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.