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Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds


Author: Yunyan Yang
Journal: Proc. Amer. Math. Soc. 136 (2008), 4095-4102
MSC (2000): Primary 58J05, 58J35
DOI: https://doi.org/10.1090/S0002-9939-08-09398-2
Published electronically: June 11, 2008
MathSciNet review: 2425752
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Abstract: Let $ (M,g)$ be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation

$\displaystyle \frac{\partial u}{\partial t}=\Delta u+au\log u+bu$

on $ M\times [0,+\infty)$, where $ a$, $ b$ are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).


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Additional Information

Yunyan Yang
Affiliation: Department of Mathematics, Information School, Renmin University of China, Beijing 100872, People’s Republic of China
Email: yunyanyang@ruc.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-08-09398-2
Received by editor(s): April 19, 2007
Received by editor(s) in revised form: October 13, 2007
Published electronically: June 11, 2008
Additional Notes: The author was supported in part by the NSFC 10601065
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society

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