Unbounded Supersolutions of Some Quasilinear Parabolic Equations: a Dichotomy

TL;DR

This paper investigates the behavior of unbounded viscosity supersolutions of certain quasilinear parabolic equations, revealing a dichotomy based on their local summability properties, with applications to the p-Laplace and Porous Medium equations.

Contribution

It introduces a classification of unbounded supersolutions into two distinct classes based on local summability, providing new insights into their structure and properties.

Findings

Supersolutions are divided into two classes depending on local summability.

The dichotomy impacts understanding of solution regularity and behavior.

Applications to p-Laplace and Porous Medium equations demonstrate the classification's relevance.

Abstract

We study unbounded (viscosity) supersolutions of the Evolutionary p-Laplace Equation in the slow diffusion case. The supersolutions fall into two widely different classes, depending on whether they are locally summable to the power p-2 or not. Also the Porous Medium Equation is studied.