Harnack type inequalities for the parabolic logarithmic p-Laplacian equation

Abstract

In this note, we concern with a class of doubly nonlinear operators whose prototype is

u_t − div(|u|^m−1|Du|^p−2Du) = 0,        p > 1,         m + p = 2.

In the last few years many progresses were made in understanding the right form of the Harnack inequalities for singular parabolic equations. For doubly nonlinear equations the singular case corresponds to the range m+p < 3. For 3−p/N < m+p < 3, where N denotes the space dimension, intrinsic Harnack estimates hold. In the range 2 < m + p ≤ 3 − p/N only a weaker Harnack form survives. In the limiting case m+p = 2, only the case p = 2 was studied. In this paper we fill this gap and we study the behaviour of the solutions in the full range p > 1 and m = 2 − p.