Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations

Abstract

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold. Mathematics Subject Classification (2010): 35K65 (primary); 35B65, 35B45 (secondary). 1. Main results Let E be an open set in RN and for T > 0 let ET = E × (0, T ]. Let u be a weak solution u ∈ Cloc ( 0, T ; Lloc(E) ) ∩ L p loc(0, T ; W 1,p loc (E)) 1 < p < 2 (1.1) of a quasi-linear, singular parabolic equation of the type ut − div A(x, t, u, Du) = B(x, t, u, Du) weakly in ET (1.2) where the functions A : ET × RN+1 → RN and B : ET × RN+1 → R are only assumed to be measurable and subject to the structure conditions   A(x, t, u, Du) · Du ≥ Co|Du|p − C p |A(x, t, u, Du)| ≤ C1|Du|p−1 + C p−1 |B(x, t, u, Du)| ≤ C |Du|p−1 + C p a.e. in ET (1.3) This work has been partially supported by I.M.A.T.I. – C.N.R. – Pavia. DiBenedetto’s work partially supported by National Science Foundation grant DMS-0652385. Received March 9, 2009; accepted in revised form May 31, 2009.