Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

Kristian Moring,Juha Kinnunen,Ratan Kr. Giri

doi:10.1007/s00030-021-00694-8

Kristian Moring, Juha Kinnunen + Show 1 more

Open Access

https://doi.org/10.1007/s00030-021-00694-8

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Abstract

We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for pge 2, but little is known in the fast diffusion case 1<p<2. Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range 1<p<infty . Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case frac{2n}{n+1}<p<2. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case 1<ple frac{2n}{n+1} and the theory is not yet well understood.

Highlights

This paper studies classes of supersolutions to the parabolic p-Laplace equation∂tu − div |∇u|p−2∇u = 0. (1.1)The general theory covers the entire parameter range 1 < p < ∞, but different phenomena occur in the slow diffusion case p > 2 and in the fast diffusion case 1 < p < 2
Kinnunen and Lindqvist [15] proved that bounded p-supercaloric functions belong to the appropriate Sobolev space and are weak supersolutions to (1.1) for p ≥ 2
In this paper we show that bounded p-supercaloric functions are weak solutions to (1.1) for the entire range 1 < p < ∞

Summary

Introduction

This paper studies classes of supersolutions to the parabolic p-Laplace equation. ∂tu − div |∇u|p−2∇u = 0. This paper studies classes of supersolutions to the parabolic p-Laplace equation. The general theory covers the entire parameter range 1 < p < ∞, but different phenomena occur in the slow diffusion case p > 2 and in the fast diffusion case 1 < p < 2. For p = 2 we have the heat equation. We do consider weak solutions, and weak supersolutions and, more generally, p-supercaloric functions to (1.1). They are pointwise defined lower semicontinuous functions, finite in a dense subset, and are required to satisfy the comparison principle

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Weak supersolutions and supercaloric functions

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Barenblatt solutions

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Infinite point source solutions

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