Intrinsic Harnack estimates estimates for quasi linear singular parabolic differential equations

ntrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations are established, including the prototype p-Laplacian equation. For p in the supercritical range 2N/(N + 1) < p < 2, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in an elliptic form at fixed time. These estimates fail for the heat equation (p → 2). It is shown by counterexamples that they fail for p in the subcritical range 1 < p ≤ 2N/(N + 1). Thus the indicated supercritical 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 equations including the porous medium equation and its quasi-linear versions.