$1$-Dimensional Harnack Estimates

TL;DR

This paper establishes a Harnack-type estimate for non-negative super-solutions of 1D singular p-Laplacian parabolic equations, showing how positivity at a point influences decay behavior over space and time.

Contribution

It introduces a quantitative sidewise spreading of positivity result for singular parabolic equations, using geometric methods to derive a new form of Harnack inequality.

Findings

Super-solutions exhibit power-like decay in space.

Positivity at a point implies decay bounds over a time interval.

The decay order depends on the parameter p.

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.