{
  "id": "1012.3218",
  "version": "v1",
  "published": "2010-12-15T03:23:25.000Z",
  "updated": "2010-12-15T03:23:25.000Z",
  "title": "Convergence of the Dirichlet solutions of the very fast diffusion equation",
  "authors": [
    "Kin Ming Hui",
    "Sunghoon Kim"
  ],
  "comment": "33 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For any $-1<m<0$, $\\mu>0$, $0\\le u_0\\in L^{\\infty}(R)$ such that $u_0(x)\\le (\\mu_0 |m||x|)^{\\frac{1}{m}}$ for any $|x|\\ge R_0$ and some constants $R_0>1$ and $0<\\mu_0\\leq \\mu$, and $f,\\,g \\in C([0,\\infty))$ such that $f(t),\\, g(t) \\geq \\mu_0$ on $[0,\\infty)$ we prove that as $R\\to\\infty$ the solution $u^R$ of the Dirichlet problem $u_t=(u^m/m)_{xx}$ in $(-R,R)\\times (0,\\infty)$, $u(R,t)=(f(t)|m|R)^{1/m}$, $u(-R,t)=(g(t)|m|R)^{1/m}$ for all $t>0$, $u(x,0)=u_0(x)$ in $(-R,R)$, converges uniformly on every compact subsets of $R\\times (0,T)$ to the solution of the equation $u_t=(u^m/m)_{xx}$ in $R\\times (0,\\infty)$, $u(x,0)=u_0(x)$ in $R$, which satisfies $\\int_Ru(x,t)\\,dx=\\int_Ru_0dx-\\int_0^t(f(s)+g(s))\\,ds$ for all $0<t<T$ where $\\int_0^T(f+g)\\,ds=\\int_Ru_0dx$. We also prove that the solution constructed is equal to the solution constructed in [Hu3] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-15T03:23:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35K15",
      "35K65"
    ],
    "keywords": [
      "fast diffusion equation",
      "dirichlet solutions",
      "convergence",
      "dirichlet problem",
      "compact subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3218H"
    }
  }
}