A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic partial differential equations
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- by Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri PDF
- Proc. Amer. Math. Soc. 138 (2010), 3521-3529 Request permission
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.References
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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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2661552