Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

Abstract

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, ∂ t u = Δ p u , with 1 < p < 2 . Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of R n × [ 0 , T ] . We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1 < p ⩽ 2 n / ( n + 1 ) . The boundedness results may be also extended to the limit case p = 1 , while the positivity estimates cannot. We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1 < p < 2 , and point out their main properties. We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely ∂ t u ∈ L loc 2 .