Gradient estimates for a simple nonlinear heat equation on manifolds

Abstract

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem

on the compact Riemannian manifold (M, g) of dimension n and with non-negative Ricci curvature. Here is a constant, V is a smooth function on M with 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.

Keywords:

• Positive solution
• nonlinear heat equation
• gradient estimate

AMS Subject Classifications:

• 53C21
• 35J60

Notes

The author(s) declare(s) that there isno conflict of interests regarding the publication of this article.

Additional information

Funding

The research is partially supported by the National Natural Science Foundation of China [grant number 11271111].