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

 

A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic partial differential equations


Authors: Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri
Journal: Proc. Amer. Math. Soc. 138 (2010), 3521-3529
MSC (2010): Primary 35K65, 35K67, 35B65; Secondary 35B45
Published electronically: June 3, 2010
MathSciNet review: 2661552
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Abstract: Let $ u$ be a non-negative solution to a singular parabolic equation of $ p$-Laplacian type ($ 1<p<2$) or porous-medium type ($ 0<m<1$). If $ u$ is bounded below on a ball $ B_\rho$ by a positive number $ M$, for times comparable to $ \rho$ and $ M$, then it is bounded below by $ \sigma M$, for some $ \sigma\in(0,1)$, on a larger ball, say $ B_{2\rho}$, for comparable times. This fact, stated quantitatively in this paper, is referred to as the ``spreading of positivity'' of solutions of such singular equations and is at the heart of any form of Harnack inequality. The proof of such a ``spreading of positivity'' effect, first given in 1992, is rather involved and not intuitive. Here we give a new proof, which is more direct, being based on geometrical ideas.


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Additional Information

Emmanuele DiBenedetto
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240
Email: em.diben@vanderbilt.edu

Ugo Gianazza
Affiliation: Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
Email: gianazza@imati.cnr.it

Vincenzo Vespri
Affiliation: Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Email: vespri@math.unifi.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10525-7
PII: S 0002-9939(2010)10525-7
Received by editor(s): October 19, 2009
Published electronically: June 3, 2010
Additional Notes: The first author was supported in part by NSF grant #DMS-0652385.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.