{
  "id": "2007.06830",
  "version": "v1",
  "published": "2020-07-14T05:44:30.000Z",
  "updated": "2020-07-14T05:44:30.000Z",
  "title": "Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space",
  "authors": [
    "Kin Ming Hui",
    "Jinwan Park"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence, uniqueness, and asymptotic behaviour of the singular solution of the fast diffusion equation, which blows up at the origin for all time. For $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $\\beta<0$ and $\\alpha=\\frac{2\\beta}{1-m}$, we prove the existence and asymptotic behaviour of singular eternal self-similar solution of the fast diffusion equation. As a consequence, we prove the existence and uniqueness of solution of Cauchy problem for the fast diffusion equation. For $n=3, 4$ and $\\frac{n-2}{n+2}\\le m<\\frac{n-2}{n}$, under appropriate condition on the initial value $u_0$, we prove the asymptotic large time behaviour, the rescaled function $\\tilde u(x,t):=e^{\\alpha t}u(e^{\\beta t}x,t)$ converges uniformly on every compact subset of $\\mathbb R^n\\setminus\\{0\\}$ to $f_{\\lambda_0}$ as $t\\to\\infty$, for some $\\lambda_0>0$. Furthermore, for the radially symmetric initial value $u_0$, $3 \\le n < 8$, $1- \\sqrt{\\frac{2}{n}} \\le m \\le \\min \\left \\{\\frac{2(n-2)}{3n}, \\frac{n-2}{n+2}\\right \\}$, we also have the asymptotic large time behaviour.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-14T05:44:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fast diffusion equation",
      "asymptotic behaviour",
      "punctured euclidean space",
      "singular solution",
      "asymptotic large time behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}