{
  "id": "2208.13569",
  "version": "v1",
  "published": "2022-08-29T12:58:32.000Z",
  "updated": "2022-08-29T12:58:32.000Z",
  "title": "Asymptotic behaviour of the finite blow-up points solutions of the fast diffusion equation",
  "authors": [
    "Shu-Yu Hsu"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $n\\ge 3$, $0<m<\\frac{n-2}{n}$, $\\Omega\\subset\\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\\dots,a_{i_0}\\in\\Omega$, $\\hat{\\Omega}=\\Omega\\setminus\\{a_1,a_2,\\dots,a_{i_0}\\}$, $0\\le f\\in L^{\\infty}(\\partial\\Omega)$ and $0\\le u_0\\in L_{loc}^p(\\hat{\\Omega})$ for some constant $p>\\frac{n(1-m)}{2}$ which satisfies $\\lambda_i|x-a_i|^{-\\gamma_i}\\le u_0(x)\\le \\lambda_i'|x-a_i|^{-\\gamma_i'}\\,\\,\\forall 0<|x-a_i|<\\delta$, $i=1,\\dots, i_0$ where $\\delta>0$, $\\lambda_i'\\ge\\lambda_i>0$ and $\\frac{2}{1-m}<\\gamma_i\\le\\gamma_i<\\frac{n-2}{m}$ $\\forall i=1,2,\\dots, i_0$ are constants. We will prove the asymptotic behaviour of the finite blow-up points solution $u$ of $u_t=\\Delta u^m$ in $\\hat{\\Omega}\\times (0,\\infty)$, $u(a_i,t)=\\infty\\,\\,\\forall i=1,\\dots,i_0, t>0$, $u(x,0)=u_0(x)$ in $\\hat{\\Omega}$ and $u=f$ on $\\partial\\Omega\\times (0,\\infty)$, as $t\\to\\infty$. We will construct finite blow-up points solution in bounded cylindrical domain with appropriate lateral boundary value such that the finite blow-up points solution will oscillate between two given harmonic functions as $t\\to\\infty$. We will also prove the existence of the minimal solution of $u_t=\\Delta u^m$ in $\\hat{\\Omega}\\times (0,\\infty)$, $u(x,0)=u_0(x)$ in $\\hat{\\Omega}$, $u(a_i,t)=\\infty\\quad\\forall t>0, i=1,2\\dots,i_0$ and $u=\\infty$ on $\\partial{\\Omega}\\times (0,\\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-29T12:58:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K65",
      "35B51",
      "35B40"
    ],
    "keywords": [
      "fast diffusion equation",
      "asymptotic behaviour",
      "construct finite blow-up points solution",
      "appropriate lateral boundary value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}