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arXiv · 1503.07448

$1$-Dimensional Harnack Estimates

Fatma Gamze Düzgün·Ugo Gianazza·Vincenzo Vespri
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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 1.1, 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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Fatma Gamze Düzgün, Ugo Gianazza, Vincenzo Vespri. 2015-03-25. $1$-Dimensional Harnack Estimates. https://doi.org/10.3934/dcdss.2016021

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