{
  "id": "0810.2521",
  "version": "v1",
  "published": "2008-10-14T19:08:57.000Z",
  "updated": "2008-10-14T19:08:57.000Z",
  "title": "Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process",
  "authors": [
    "Liu Qilin",
    "Liang Fei",
    "Li Yuxiang"
  ],
  "comment": "20pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \\[ u_{t}=\\Delta u+\\displaystyle\\frac{\\lambda f(u)}{\\big(\\int_{\\Omega}f(u)dx\\big)^{p}}, x\\in \\Omega, t>0, \\] with homogeneous Dirichlet boundary condition, where $\\lambda>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\\leq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $\\lambda>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $\\lambda>0$; (c) For $p=2$, if $0<\\lambda<2|\\partial\\Omega|^2$, then $u(x,t)$ is globally bounded, if $\\lambda=2|\\partial\\Omega|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)\\to\\infty$ as $t\\to\\infty$ for all $x\\in\\Omega$, if $\\lambda>2|\\partial\\Omega|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $x\\in\\Omega$; (d) For $p>2$, there exists a $\\lambda^*>0$ such that for $\\lambda>\\lambda^*$, or for $0<\\lambda\\leq\\lambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $p\\geq2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-14T19:08:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B40",
      "35K60"
    ],
    "keywords": [
      "nonlocal parabolic problem",
      "ohmic heating process",
      "asymptotic behavior",
      "finite time",
      "homogeneous dirichlet boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.2521Q"
    }
  }
}