Subpotential lower bounds for nonnegative solutions to certain quasi-linear degenerate parabolic equations

Abstract

Nonnegative weak solutions of quasi-linear degenerate parabolic equations of $p$-Laplacian type are shown to be locally bounded below by Barenblatt-type subpotentials. As a consequence, nonnegative solutions expand their positivity set. That is, a quantitative lower bound on a ball $B_{\rho }$ at time $\stackrel{̲}{t}$ yields a quantitative lower bound on a ball $B_{2\rho }$ at some further time $t$. These lower bounds also permit one to recast the Harnack inequality of [4] in a family of alternative, equivalent forms