Gradient estimates for a simple nonlinear heat equation on manifolds

TL;DR

This paper derives gradient estimates for positive solutions to a nonlinear heat equation on compact Riemannian manifolds with non-negative Ricci curvature, linking the equation to Perelman's W-functional and Log-Sobolev inequalities.

Contribution

It provides new gradient estimates for a nonlinear heat equation related to Perelman's W-functional on manifolds with non-negative Ricci curvature.

Findings

Established gradient bounds for solutions to the nonlinear heat equation.

Connected the heat flow to Log-Sobolev inequalities and scalar curvature.

Extended understanding of heat equations on curved manifolds.

Abstract

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $$ u_t-\Delta u=au\log u+Vu, \ \ u>0 $$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-\Delta V\leq A$ for some positive constant $A$. This heat equation is a basic evolution equation and it can be considered as the negative gradient heat flow to $W$-functional (introduced by G.Perelman), which is the Log-Sobolev inequalities on the Riemannian manifold and $V$ corresponds to the scalar curvature.