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

Funding Information: The authors would like to thank Peter Lindqvist for useful discussions and the Academy of Finland for support. K. Moring has also been supported by the Magnus Ehrnrooth Foundation. Funding Information: The authors would like to thank Peter Lindqvist for useful discussions and the Academy of Finland for support. K. Moring has also been supported by the Magnus Ehrnrooth Foundation. Publisher Copyright: © 2021, The Author(s). Copyright: Copyright 2021 Elsevier B.V., All rights reserved.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 p≥ 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< ∞. 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 2nn+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<p≤2nn+1 and the theory is not yet well understood.Peer reviewe