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

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, ∂tu = ∆pu, with 1 < p < 2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of R× [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 ≤ 2n/(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 ∂tu ∈ Lloc. AMS Subject Classification: 35B35, 35B65, 35K55, 35K65.