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, ${\partial }_{t}u={\Delta }_{p}u$, with $1<p<2$. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of $\RR^n\times [0,T]$. 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 $1<p\le 2n/(n+1)$. 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 ${\partial }_{t}u\in L_{\mathrm{loc}}^2$.