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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
DOI: https://doi.org/10.1090/S0002-9939-2010-10525-7
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
ORCID: 0000-0003-2558-560X
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

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.