TY - JOUR
T1 - Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case
AU - Giri, Ratan Kr
AU - Kinnunen, Juha
AU - Moring, Kristian
N1 - 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.
PY - 2021/5
Y1 - 2021/5
N2 - 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
AB - 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
KW - Comparison principle
KW - Moser iteration
KW - Obstacle problem
KW - p-supercaloric function
KW - Parabolic p-Laplace equation
UR - http://www.scopus.com/inward/record.url?scp=85104547093&partnerID=8YFLogxK
U2 - 10.1007/s00030-021-00694-8
DO - 10.1007/s00030-021-00694-8
M3 - Article
AN - SCOPUS:85104547093
SN - 1021-9722
VL - 28
JO - Nonlinear Differential Equations and Applications
JF - Nonlinear Differential Equations and Applications
IS - 3
M1 - 33
ER -