Sharp gradient estimates for a heat equation in Riemannian manifolds

TL;DR

This paper establishes optimal gradient estimates for positive solutions to a heat equation on Riemannian manifolds, leading to a Liouville-type result under sublinear growth conditions, improving previous bounds.

Contribution

It provides sharp gradient estimates for a nonlinear heat equation on Riemannian manifolds, enhancing existing results and deriving a new Liouville-type theorem.

Findings

Sharp gradient estimates for the heat equation with a logarithmic term.

Positive solutions with sublinear growth are constant.

Improved bounds over previous works.

Abstract

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=\Delta u+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the equation $u_t=\Delta u$ and $\log u$ is of sublinear growth in both spatial and time directions then $u$ must be constant. This gradient estimate is sharp since it is well-known that $u(x,t)=e^{x+t}$ satisfying $u_t=\Delta u$. We also emphasize that our results are better than those given by Jiang (\cite{XJ16}), Souplet-Zhang (\cite{SZ06}), Wu (\cite{Wu15, Wu17}), and others.