{
  "id": "math/0609513",
  "version": "v2",
  "published": "2006-09-19T15:02:08.000Z",
  "updated": "2007-05-02T17:38:04.000Z",
  "title": "On the extinction profile of solutions to fast-diffusion",
  "authors": [
    "Panagiota Daskalopoulos",
    "Natasa Sesum"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the extinction behavior of solutions to the fast diffusion equation $u_t = \\Delta u^m$ on $\\R^N\\times (0,T)$, in the range of exponents $m \\in (0, \\frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in between two Barenblatt solutions vanishing at time $T$, then the vanishing behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions $u$ with initial data $u_0 = B (1 + o(1))$, near $ |x| >> 1$, which live longer than $B$ and change behaviour at $T$. The behavior of such solutions is governed by $B(\\cdot,t)$ up to $T$, while for $t >T$ the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow ($m = \\frac{N-2}{N+2}$) the above means that these solutions $u$ develop a singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$ they immediately smoothen up and exhibit the vanishing profile of a sphere. In the appendix we show how to remove the assumption on the bound on $u_0$ from below by a Barenblatt.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-05-02T17:38:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40"
    ],
    "keywords": [
      "extinction profile",
      "initial data",
      "fast-diffusion",
      "fast diffusion equation",
      "barenblatt solution disappears"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9513D"
    }
  }
}