{
  "id": "1407.7445",
  "version": "v1",
  "published": "2014-07-28T15:28:24.000Z",
  "updated": "2014-07-28T15:28:24.000Z",
  "title": "Global smooth solutions of 3-D quasilinear wave equations with small initial data",
  "authors": [
    "Ding Bingbing",
    "Liu Yingbo",
    "Yin Huicheng"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we are concerned with the 3-D quasilinear wave equation $ \\ds\\sum_{i,j=0}^3g^{ij}(u, \\p u)\\p_{ij}^2u$ $=0$ with $(u(0,x), \\p_tu(0,x))=(\\ve u_0(x), \\ve u_1(x))$, where $x_0=t$, $x=(x_1, x_2, x_3)$, $\\p=(\\p_0, \\p_1, ..., \\p_3)$, $u_0(x), u_1(x)\\in C_0^\\infty(\\Bbb R^3)$, $\\ve>0$ is small enough, and $g^{ij}(u, \\p u)=g^{ji}(u, \\p u)$ are smooth in their arguments. Without loss of generality, one can write $g^{ij}(u, \\p u)=c^{ij}+d^{ij}u+\\ds\\sum_{k=0}^3e^{ij}_k\\p_ku+O(|u|^2+|\\p u|^2)$, where $c^{ij}, d^{ij}$ and $e^{ij}_k$ are some constants, and $\\ds\\sum_{i,j=0}^3c^{ij}\\p_{ij}^2=-\\square\\equiv -\\p_t^2+\\Delta$. When $\\ds\\sum_{i,j,k=0}^3e^{ij}_k\\o_k\\o_i\\o_j\\not\\equiv 0$ for $\\o_0=-1$ and $\\o=(\\o_1, \\o_2, \\o_3)\\in\\Bbb S^2$, the authors in [7-8] have shown the blowup of the smooth solution $u$ in finite time as long as $(u_0(x), u_1(x))\\not\\equiv 0$. In the present paper, when $\\ds\\sum_{i,j,k=0}^3e^{ij}_k\\o_k\\o_i\\o_j\\equiv 0$, we will prove the global existence of the smooth solution $u$. Therefore, the complete results on the blowup or global existence of the small data solutions have been established for the general 3-D quasilinear wave equations $\\ds\\sum_{i,j=0}^3g^{ij}(u, \\p u)\\p_{ij}^2u=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-28T15:28:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35L72"
    ],
    "keywords": [
      "quasilinear wave equation",
      "small initial data",
      "global smooth solutions",
      "global existence",
      "small data solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7445B"
    }
  }
}