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

Abstract:

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.