{
  "id": "0805.4823",
  "version": "v2",
  "published": "2008-05-30T18:39:47.000Z",
  "updated": "2008-12-01T14:50:20.000Z",
  "title": "Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations",
  "authors": [
    "Matteo Bonforte",
    "Juan Luis Vazquez"
  ],
  "comment": "36 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate qualitative properties of local solutions $u(t,x)\\ge 0$ to the fast diffusion equation, $\\partial_t u =\\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\\times\\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\\le 0$, while the positivity ones cannot be true in that range.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-12-01T14:50:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35B65",
      "35K55",
      "35K65"
    ],
    "keywords": [
      "fast diffusion equation",
      "harnack inequalities",
      "local smoothing",
      "positivity",
      "general nonnegative initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.4823B"
    }
  }
}