Sharp gradient estimates for a heat equation in Riemannian manifolds

2018/10/07 by Dung, Ha Tuan, Dung, Nguyen Thac · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.03189

Abstract

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u_t=Δu+aulog u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation u_t=Δu 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=Δu. We also emphasize that our results are better than those given by Jiang (\citeXJ16), Souplet-Zhang (\citeSZ06), Wu (\citeWu15, Wu17), and others.

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• Optimal Hamilton-type gradient estimates and large time heat kernel bounds for noncompact manifolds

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