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arXiv · 1810.03189

Sharp gradient estimates for a heat equation in Riemannian manifolds

Ha Tuan Dung·Nguyen Thac Dung
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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.

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BibTeXRIS

Ha Tuan Dung, Nguyen Thac Dung. 2018-10-07. Sharp gradient estimates for a heat equation in Riemannian manifolds. https://arxiv.org/abs/1810.03189

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