Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations

Abstract

Intrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations, are established, including the prototype $p$-Laplacean equation (1.4) below. For $p$ in the super-critical range $\frac{2N}{N+1}<p<2$, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in a elliptic form at fixed time. These estimates fail for the heat equation ($p\to 2$). It is shown by counterexamples, that they fail for $p$ in the sub-critical range $1<p\le \frac{2N}{N+1}$. Thus the indicated super-critical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure theoretical arguments, as opposed to comparison principles and are sufficiently flexible to hold for a large class of singular parabolic equation including the porous medium equation and its quasi-linear versions.