$1$-dimensional Harnack estimates

Fatma Gamze  Düzgün; Ugo Gianazza; Vincenzo  Vespri

doi:10.3934/dcdss.2016021

Article Contents

2016, Volume 9, Issue 3: 675-685. Doi: 10.3934/dcdss.2016021

This issue Previous Article A singular limit problem for the Ibragimov-Shabat equation Next Article Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel

$1$-dimensional Harnack estimates

1.
Hacettepe University, 06800, Beytepe, Ankara
2.
Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia
3.
Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, viale Morgagni, 67/A, 50134, Firenze

Received: March 2015

Revised: July 2015

Published: June 2016

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Abstract

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1< p <2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.2, is a ``sidewise spreading of positivity'' of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.

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Mathematics Subject Classification: Primary: 35K65, 35B65; Secondary: 35B45.

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References

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