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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: 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

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.

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