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

• Published in: Nonlinear differential equations and applications 2021-05, Vol.28 (3), Article 33
• Main Authors: Giri, Ratan Kr, Kinnunen, Juha, Moring, Kristian
• Format: Article
• Language: English
• Subjects: Analysis; Diffusion rate; Laplace equation; Mathematics; Mathematics and Statistics; Sobolev space
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-021-00694-8

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 2 n 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 < p ≤ 2 n n + 1 and the theory is not yet well understood.