# Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

## TL;DR

This paper derives gradient estimates and Harnack inequalities for solutions of a parabolic equation on a manifold evolving under a geometric flow, providing tools for analyzing such equations in geometric analysis.

## Contribution

It introduces new gradient estimates and Harnack inequalities for parabolic equations on manifolds under general geometric flows, extending previous results to more general settings.

## Key findings

- Established space-time gradient estimates for positive solutions.
- Derived elliptic type gradient estimates for bounded positive solutions.
- Obtained Harnack inequalities from the gradient estimates.

## Abstract

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (\Delta -q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.