Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case
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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Date
2021-05
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Language
en
Pages
21
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Nonlinear Differential Equations and Applications, Volume 28, issue 3
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 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.Description
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.
Keywords
Comparison principle, Moser iteration, Obstacle problem, p-supercaloric function, Parabolic p-Laplace equation
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Citation
Giri, R K, Kinnunen, J & Moring, K 2021, ' Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case ', Nonlinear Differential Equations and Applications, vol. 28, no. 3, 33 . https://doi.org/10.1007/s00030-021-00694-8