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Sharp gradient estimates for a heat equation in Riemannian manifolds

Authors: Ha Tuan Dung and Nguyen Thac Dung
Journal: Proc. Amer. Math. Soc. 147 (2019), 5329-5338
MSC (2010): Primary 32M05; Secondary 32H02
Published electronically: July 8, 2019
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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$.

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Ha Tuan Dung
Affiliation: Faculty of Mathematics, Hanoi Pedagogical University No. 2, Xuan Hoa, Vinh Phuc, Vietnam; and Department of Mathematics, National Tsing Hua University, Hsin-Chu, Taiwan

Nguyen Thac Dung
Affiliation: Faculty of Mathematics - Mechanics - Informatics, Hanoi University of Science (VNU), Hanoi, Vietnam; and Thang Long Institute of Mathematics and Applied Sciences (TIMAS), Thang Long Univeristy, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam

Keywords: Ancient solution, heat equation, Liouville theorem, sharp gradient estimate, sublinear growth
Received by editor(s): November 20, 2018
Received by editor(s) in revised form: March 29, 2019
Published electronically: July 8, 2019
Additional Notes: The first author was supported by the Research Fund of Hanoi Pedagogical University No. 2 (Vietnam) under grant number C.2019.05.
The second author was partially supported by NAFOSTED under grant number 101.02-2017.313.
Communicated by: Guofang Wei
Article copyright: © Copyright 2019 American Mathematical Society