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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

A new result for the porous medium equation derived from the Ricci flow


Author: Lang-Fang Wu
Journal: Bull. Amer. Math. Soc. 28 (1993), 90-94
MSC: Primary 58G30; Secondary 35Q51, 53C21, 58G11, 76S05
MathSciNet review: 1164949
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Abstract: Given $ {\mathbb{R}^2}$, with a "good" complete metric, we show that the unique solution of the Ricci flow approaches a soliton at time infinity. Solitons are solutions of the Ricci flow, which move only by diffeomorphism. The Ricci flow on $ {\mathbb{R}^2}$ is the limiting case of the porous medium equation when m is zero. The results in the Ricci flow may therefore be interpreted as sufficient conditions on the initial data, which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a "soliton-solution".


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DOI: http://dx.doi.org/10.1090/S0273-0979-1993-00336-7
PII: S 0273-0979(1993)00336-7
Article copyright: © Copyright 1993 American Mathematical Society