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

TL;DR

This paper investigates the properties of solutions to the fast p-Laplacian equation for 1<p<2, establishing new positivity, boundedness, and Harnack inequalities, along with existence and asymptotic behavior of large solutions.

Contribution

It introduces new intrinsic Harnack inequalities in the very fast diffusion range and extends boundedness results to the limit case p=1, providing a comprehensive analysis of solution properties.

Findings

Quantitative positivity and boundedness estimates for solutions.

New intrinsic Harnack inequalities for 1<p≤2n/(n+1).

Existence and sharp asymptotics for large solutions.

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 $\partial_t u\in L^2_{\rm loc}$.