Supercaloric functions for the porous medium equation in the fast diffusion case

TL;DR

This paper investigates supercaloric functions related to the porous medium equation in the fast diffusion regime, establishing their properties, classifications, and connections to weak supersolutions.

Contribution

It introduces a comprehensive analysis of supercaloric functions, including their classification, characterization, and relation to weak supersolutions in the fast diffusion case.

Findings

Bounded supercaloric functions are weak supersolutions.

Unbounded supercaloric functions are classified into two classes based on the Barenblatt and infinite point-source solutions.

Connections between supercaloric functions and weak supersolutions are established.

Abstract

We study a generalized class of supersolutions, so-called supercaloric functions to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison principle. We prove that bounded supercaloric functions are weak supersolutions. In the supercritical range, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes dictated by the Barenblatt solution and the infinite point-source solution, and give several characterizations for these classes. Furthermore, we study the pointwise behavior of supercaloric functions and obtain connections between supercaloric functions and weak supersolutions.