{
  "id": "1303.4225",
  "version": "v1",
  "published": "2013-03-18T12:18:14.000Z",
  "updated": "2013-03-18T12:18:14.000Z",
  "title": "Blowup of classical solutions for a class of 3-D quasilinear wave equations with small initial data",
  "authors": [
    "Bingbing Ding",
    "Ingo Witt",
    "Huicheng Yin"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation $\\partial_t^2u-\\left (1+u+\\p_t u\\right)\\Delta u=0$. This equation is prototypical of the more general equation $\\dsize\\sum_{i,j=0}^3g_{ij}(u, \\nabla u)\\partial_{ij}u=0$, where $x_0=t$ and $g_{ij}(u, \\nabla u)=c_{ij}+d_{ij}u+\\dsize\\sum_{k=0}^3e_{ij}^k\\partial_ku+O(|u|^2+|\\nabla u|^2)$ are smooth functions of their arguments, with $c_{ij}, d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\\neq0$ for some $(i,j)$; moreover, $\\dsize\\sum_{i,j,k=0}^3e_{ij}^k(\\partial_ku)\\p_{ij} u$ does not fulfill the null condition. For the 3-D nonlinear wave equations $\\partial_t^2u-\\left (1+u\\right)\\Delta u=0$ and $\\partial_t^2u-\\left (1+\\partial_t u\\right)\\Delta u=0$, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small smooth data solutions. For radial initial data, we show that the small smooth data solution of $\\partial_t^2u-\\left(1+u+\\partial_t u\\right)\\Delta u=0$ blows up in finite time. The explicit expression of the asymptotic lifespan $T_{\\varepsilon}$ as $\\varepsilon\\to0^+$ is also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-18T12:18:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35J70",
      "35R35"
    ],
    "keywords": [
      "quasilinear wave equations",
      "small initial data",
      "small smooth data solution",
      "classical solutions",
      "nonlinear wave equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.4225D"
    }
  }
}