{
  "id": "1504.07301",
  "version": "v1",
  "published": "2015-04-27T23:14:00.000Z",
  "updated": "2015-04-27T23:14:00.000Z",
  "title": "Asymptotic behavior for a nonlocal diffusion equation in exterior domains: the critical two-dimensional case",
  "authors": [
    "Carmen Cortázar",
    "Manuel Elgueta",
    "Fernando Quirós",
    "Noemi Wolanski"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, $\\partial _t u=J*u-u$, where $J$ is a smooth, radially symmetric kernel with support $B_d(0)\\subset\\mathbb{R}^2$. The problem is set in an exterior two-dimensional domain which excludes a hole $\\mathcal{H}$, and with zero Dirichlet data on $\\mathcal{H}$. In the far field scale, $\\xi_1\\le |x|t^{-1/2}\\le \\xi_2$ with $\\xi_1,\\xi_2>0$, the scaled function $\\log t\\, u(x,t)$ behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by $J$. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic \\lq logarithmic momentum' of the solution, $\\lim_{t\\to\\infty}\\int_{\\mathbb{R}^2}u(x,t)\\log|x|\\,dx$. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, $|x|\\le t^{1/2}h(t)$ with $\\lim_{t\\to\\infty} h(t)=0$, the scaled function $t(\\log t)^2u(x,t)/\\log |x|$ converges to a multiple of $\\phi(x)/\\log |x|$, where $\\phi$ is the unique stationary solution of the problem that behaves as $\\log|x|$ when $|x|\\to\\infty$. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, $|x|\\ge t^{1/2} g(t)$ with $g(t)\\to\\infty$, the solution is proved to be of order $o((t\\log t)^{-1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-27T23:14:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R09",
      "45K05",
      "45M05"
    ],
    "keywords": [
      "nonlocal diffusion equation",
      "critical two-dimensional case",
      "asymptotic behavior",
      "exterior domains",
      "far field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150407301C"
    }
  }
}