Supercaloric functions for the porous medium equation in the fast diffusion case

• Kristian Moring ^[2] ; Christoph Scheven ^[1]
  1. [1] University of Duisburg-Essen

     University of Duisburg-Essen

     Kreisfreie Stadt Essen, Alemania

     
  2. [2] Paris-Lodron-Universität Salzburg, Salzburg, Austria
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 41, Nº 2, 2025, págs. 509-550
• Idioma: inglés
• DOI: 10.4171/RMI/1507
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• Resumen

  • We study a generalized class of supersolutions, so-called supercaloric functions, to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison principle. We prove that bounded supercaloric functions are weak supersolutions. In the supercritical range, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes dictated by the Barenblatt solution and the infinite point-source solution, and give several characterizations for these classes. Furthermore, we study the pointwise behavior of supercaloric functions and obtain connections between supercaloric functions and weak supersolutions.