Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds
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- by Xiaobao Zhu
- Proc. Amer. Math. Soc. 139 (2011), 1637-1644
- DOI: https://doi.org/10.1090/S0002-9939-2010-10824-9
- Published electronically: December 13, 2010
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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 \begin{align*} \partial _{t}u=\Delta u^{\alpha },\ \ 1-\frac {2}{n}<\alpha <1 \end{align*} on $M\times (-\infty ,0]$. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.References
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Bibliographic 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
- Received by editor(s): May 8, 2010
- Published electronically: December 13, 2010
- Communicated by: Chuu-Lian Terng
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1637-1644
- MSC (2010): Primary 35B45, 35B53, 35K55, 35K65, 58J35
- DOI: https://doi.org/10.1090/S0002-9939-2010-10824-9
- MathSciNet review: 2763753