{
  "id": "1103.5257",
  "version": "v1",
  "published": "2011-03-27T23:20:38.000Z",
  "updated": "2011-03-27T23:20:38.000Z",
  "title": "Smooth solutions to the nonlinear wave equation can blow up on Cantor sets",
  "authors": [
    "Rowan Killip",
    "Monica Visan"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct $C^\\infty$ solutions to the one-dimensional nonlinear wave equation $$ u_{tt} - u_{xx} - \\tfrac{2(p+2)}{p^2} |u|^p u=0 \\quad \\text{with} \\quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\\infty$ hypersurface. As a corollary, we show that smooth solutions can blow up (at the first instant) on an arbitrary compact set. We also construct solutions that blow up on general space-like $C^k$ hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-27T23:20:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth solutions",
      "cantor sets",
      "one-dimensional nonlinear wave equation",
      "arbitrary compact set",
      "construct solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.5257K"
    }
  }
}