Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

TL;DR

This paper establishes new positivity, boundedness, and Harnack inequalities for solutions to the very fast diffusion equation, especially for low values of the parameter m, extending understanding of the equation's qualitative behavior.

Contribution

It provides the first quantitative positivity and boundedness estimates, along with new Harnack inequalities, for very fast diffusion equations with low m, including the case m ≤ 0.

Findings

New positivity estimates for solutions in the very fast diffusion range.

Boundedness results valid even for m ≤ 0.

Harnack inequalities characteristic of fast diffusion equations.

Abstract

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\times\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0$, while the positivity ones cannot be true in that range.