{
  "id": "1606.03793",
  "version": "v1",
  "published": "2016-06-13T02:07:25.000Z",
  "updated": "2016-06-13T02:07:25.000Z",
  "title": "Singular limits and properties of solutions of some degenerate elliptic and parabolic equations",
  "authors": [
    "Kin Ming Hui",
    "Sunghoon Kim"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $n\\geq 3$, $0\\le m<\\frac{n-2}{n}$, $\\rho_1>0$, $\\beta>\\beta_0^{(m)}=\\frac{m\\rho_1}{n-2-nm}$, $\\alpha_m=\\frac{2\\beta+\\rho_1}{1-m}$ and $\\alpha=2\\beta+\\rho_1$. For any $\\lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\\La(v^m/m)+\\alpha_m v+\\beta x\\cdot\\nabla v=0$, $v>0$, in $\\R^n\\setminus\\{0\\}$ which satisfies $\\lim_{|x|\\to 0}|x|^{\\frac{\\alpha_m}{\\beta}}v^{(m)}(x)=\\lambda^{-\\frac{\\rho_1}{(1-m)\\beta}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $m\\to 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $\\R^n\\setminus\\{0\\}$ to the solution $v$ of $\\La\\log v+\\alpha v+\\beta x\\cdot\\nabla v=0$, $v>0$, in $\\R^n\\bs\\{0\\}$, which satisfies $\\lim_{|x|\\to 0}|x|^{\\frac{\\alpha}{\\beta}}v(x)=\\lambda^{-\\frac{\\rho_1}{\\beta}}$. We also prove that if the solution $u^{(m)}$ of $u_t=\\Delta (u^m/m)$, $u>0$, in $(\\R^n\\setminus\\{0\\})\\times (0,T)$ which blows up near $\\{0\\}\\times (0,T)$ at the rate $|x|^{-\\frac{\\alpha_m}{\\beta}}$ satisfies some mild growth condition on $(\\R^n\\setminus\\{0\\})\\times (0,T)$, then as $m\\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2,1}(K)$ for any compact subset $K$ of $(\\R^n\\setminus\\{0\\})\\times (0,T)$ to the solution of $u_t=\\La\\log u$, $u>0$, in $(\\R^n\\setminus\\{0\\})\\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\\La \\log v+\\alpha v+\\beta x\\cdot\\nabla v=0$, $v>0$, in $\\R^n\\setminus\\{0\\}$, which satisfies $\\lim_{|x|\\to 0}|x|^{\\frac{\\alpha}{\\beta}}v(x)=\\lambda^{-\\frac{\\rho_1}{\\beta}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-13T02:07:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic equations",
      "singular limits",
      "degenerate elliptic",
      "compact subset",
      "properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}