{
  "id": "1710.06108",
  "version": "v1",
  "published": "2017-10-17T05:54:08.000Z",
  "updated": "2017-10-17T05:54:08.000Z",
  "title": "Global behaviour of solutions of the fast diffusion equation",
  "authors": [
    "Shu-Yu Hsu"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We will extend a recent result of B.~Choi and P.~Daskalopoulos (\\cite{CD}). For any $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $m\\ne\\frac{n-2}{n+2}$, $\\beta>0$ and $\\lambda>0$, we prove the higher order expansion of the radially symmetric solution $v_{\\lambda,\\beta}(r)$ of $\\frac{n-1}{m}\\Delta v^m+\\frac{2\\beta}{1-m} v+\\beta x\\cdot\\nabla v=0$ in $\\mathbb{R}^n$, $v(0)=\\lambda$, as $r\\to\\infty$. As a consequence for any $n\\ge 3$ and $0<m<\\frac{n-2}{n}$ if $u$ is the solution of the equation $u_t=\\frac{n-1}{m}\\Delta u^m$ in $\\mathbb{R}^n\\times (0,\\infty)$ with initial value $0\\le u_0\\in L^{\\infty}(\\mathbb{R}^n)$ satisfying $u_0(x)^{1-m}= \\frac{2(n-1)(n-2-nm)}{(1-m)\\beta |x|^2}\\left(\\log |x|-\\frac{n-2-(n+2)m}{2(n-2-nm)}\\log (\\log |x|)+K_1+o(1))\\right)$ as $|x|\\to\\infty$ for some constants $\\beta>0$ and $K_1\\in\\mathbb{R}$, then as $t\\to\\infty$ the rescaled function $\\widetilde{u}(x,t)=e^{\\frac{2\\beta}{1-m}t}u(e^{\\beta t}x,t)$ converges uniformly on every compact subsets of $\\mathbb{R}^n$ to $v_{\\lambda_1,\\beta}$ for some constant $\\lambda_1>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-17T05:54:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35B20",
      "35B09"
    ],
    "keywords": [
      "fast diffusion equation",
      "global behaviour",
      "higher order expansion",
      "compact subsets",
      "initial value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}