{
  "id": "1109.3618",
  "version": "v1",
  "published": "2011-09-16T13:56:52.000Z",
  "updated": "2011-09-16T13:56:52.000Z",
  "title": "Existence and asymptotic behaviour of solutions of the very fast diffusion equation",
  "authors": [
    "Shu-Yu Hsu"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Let n>2, $0<m\\le (n-2)/n$, p>\\max(1,(1-m)n/2), and $0\\le u_0\\in L_{loc}^p(R^n)$ satisfy $\\liminf_{R\\to\\infty}R^{-n+\\frac{2}{1-m}}\\int_{|x|\\le R}u_0\\,dx=\\infty$. We prove the existence of unique global classical solution of $u_t=\\frac{n-1}{m}\\Delta u^m$, u>0, in $R^n\\times (0,\\infty)$, u(x,0)=u_0(x) in $\\R^n$. If in addition 0<m<(n-2)/n and $u_0(x)\\approx A|x|^{-q}$ as $|x|\\to\\infty$ for some constants A>0, q<n/p, we prove that there exist constants $\\alpha$, $\\beta$, such that the function $v(x,t)=t^{\\alpha}u(t^{\\beta}x,t)$ converges uniformly on every compact subset of $R^n$ to the self-similar solution $\\psi(x,1)$ of the equation with $\\psi(x,0)=A|x|^{-q}$ as $t\\to\\infty$. Note that when m=(n-2)/(n+2), n>2, if $g_{ij}=u^{\\frac{4}{n+2}}\\delta_{ij}$ is a metric on $R^n$ that evolves by the Yamabe flow $\\partial g_{ij}/\\partial t=-Rg_{ij}$ with u(x,0)=u_0(x) in $R^n$ where $R$ is the scalar curvature, then u(x,t) is a global solution of the above fast diffusion equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-16T13:56:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K15",
      "35B40",
      "35K65",
      "58J35"
    ],
    "keywords": [
      "fast diffusion equation",
      "asymptotic behaviour",
      "unique global classical solution",
      "global solution",
      "scalar curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.3618H"
    }
  }
}