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arXiv · 2306.07155

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

Kristian Moring·Christoph Scheven
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Abstract

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.

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Kristian Moring, Christoph Scheven. 2023-06-12. Supercaloric functions for the porous medium equation in the fast diffusion case. https://arxiv.org/abs/2306.07155

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