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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds


Author: Xiaobao Zhu
Journal: Proc. Amer. Math. Soc. 139 (2011), 1637-1644
MSC (2010): Primary 35B45, 35B53, 35K55, 35K65, 58J35
Published electronically: December 13, 2010
MathSciNet review: 2763753
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a complete noncompact Riemannian manifold of dimension $ n$. In this paper, we derive a local gradient estimate for positive solutions of fast diffusion equations

$\displaystyle \partial_{t}u=\Delta u^{\alpha}, \ 1-\frac{2}{n}<\alpha<1$    

on $ M\times(-\infty,0]$. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.


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Additional Information

Xiaobao Zhu
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: zhuxiaobao@amss.ac.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10824-9
PII: S 0002-9939(2010)10824-9
Keywords: Gradient estimate, fast diffusion equation, Liouville theorem
Received by editor(s): May 8, 2010
Published electronically: December 13, 2010
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.