Intrinsic Harnack Estimates for Nonnegative Local Solutions of Degenerate Parabolic Equations

We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic p-Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear p-Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations. 1. Main results Let E be an open set in R , and for T > 0, let ET denote the cylindrical domain E × (0, T ]. Consider quasi-linear, parabolic differential equations of the form (1.1) ut − divA(x, t, u,Du) = b(x, t, u,Du) weakly in ET , where the functions A : ET × R → R and b : ET × R → R are only assumed to be measurable and subject to the structure conditions (1.2) ⎧⎨ ⎩ A(x, t, u,Du) ·Du ≥ Co|Du| − C, |A(x, t, u,Du)| ≤ C1|Du| + Cp−1, |b(x, t, u,Du)| ≤ C|Du|p−1 + Cp−1, a.e. in ET , where p ≥ 2 and Co and C1 are given positive constants, and C is a given nonnegative constant. A function (1.3) u ∈ Cloc ( 0, T ;Lloc(E) ) ∩ Lploc ( 0, T ;W 1,p loc (E) ) is a local weak solution to (1.1) if for every compact setK ⊂ E and every subinterval [t1, t2] ⊂ (0, T ],