$1$-Dimensional Harnack Estimates

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 $\mathbb{R}\times [T\mathrm{/}2,T]$. 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.