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

Ratan Kr Giri, Juha Kinnunen*, Kristian Moring

*Corresponding author for this work

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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.

Original languageEnglish
Article number33
Number of pages21
JournalNonlinear Differential Equations and Applications
Issue number3
Publication statusPublished - May 2021
MoE publication typeA1 Journal article-refereed


  • Comparison principle
  • Moser iteration
  • Obstacle problem
  • p-supercaloric function
  • Parabolic p-Laplace equation


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