# Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

## TL;DR

This paper derives new gradient estimates and Harnack inequalities for positive solutions to nonlinear parabolic equations under Ricci flow, generalizing and improving previous results in the field.

## Contribution

It introduces novel gradient estimates and Harnack inequalities for nonlinear parabolic equations under Ricci flow, extending classical results to this geometric setting.

## Key findings

- Gradient estimates for solutions under Ricci flow
- Harnack inequalities for positive solutions
- Generalization of Li-Yau and Hamilton estimates

## Abstract

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gradient estimates and Harnack inequalities for positive solutions to the heat equation under Ricci flow are derived. These results in the paper can be regard as generalizing the gradient estimates of Li-Yau, J. Y. Li, Hamilton and Li-Xu to the Ricci flow. Our results also improve the estimates of S. P. Liu and J. Sun to the nonlinear parabolic equation under Ricci flow.