{
  "id": "2008.12540",
  "version": "v1",
  "published": "2020-08-28T08:59:19.000Z",
  "updated": "2020-08-28T08:59:19.000Z",
  "title": "Supercaloric functions for the parabolic $p$-Laplace equation in the fast diffusion case",
  "authors": [
    "Ratan Kr. Giri",
    "Juha Kinnunen",
    "Kristian Moring"
  ],
  "categories": [
    "math.AP"
  ],
  "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 $p\\ge 2$, 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 $1<p<\\infty$. 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 $\\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\\le \\frac{2n}{n+1}$ and the theory is not yet well understood.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-28T08:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35K67"
    ],
    "keywords": [
      "fast diffusion case",
      "laplace equation",
      "sharp local integrability estimates",
      "infinite point source solution",
      "barenblatt solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}