Harnack Estimate For Positive Solutions to a Nonlinear Equation Under Geometric Flow

TL;DR

This paper derives gradient estimates for positive solutions to a nonlinear parabolic equation evolving under a general geometric flow on complete noncompact manifolds, extending understanding of solution behavior in geometric analysis.

Contribution

It introduces new gradient estimate techniques for nonlinear equations under geometric flows on noncompact manifolds, broadening previous results.

Findings

Established gradient bounds for solutions under general geometric flows

Extended estimates to noncompact manifold settings

Provided tools for analyzing nonlinear PDEs in geometric contexts

Abstract

In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.