{
  "id": "1206.1167",
  "version": "v2",
  "published": "2012-06-06T10:07:20.000Z",
  "updated": "2013-02-22T21:10:58.000Z",
  "title": "Asymptotic behavior for the heat equation in nonhomogeneous media with critical density",
  "authors": [
    "Razvan Iagar",
    "Ariel Sánchez"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\\partial_{t}u=\\Delta u, \\quad \\hbox{in} \\ \\real^N\\times(0,\\infty). $$ The asymptotic behavior proves to have some interesting and quite striking properties. We show that there are two completely different asymptotic profiles depending on whether the initial data $u_0$ vanishes at $x=0$ or not. Moreover, in the former the results are true only for radially symmetric solutions, and we provide counterexamples to convergence to symmetric profiles in the general case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-22T21:10:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "heat equation",
      "nonhomogeneous media",
      "critical density",
      "general case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1167I"
    }
  }
}