A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic partial differential equations

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.