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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds
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by Xiaobao Zhu PDF
Proc. Amer. Math. Soc. 139 (2011), 1637-1644 Request permission

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.
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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
  • 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