{
  "id": "1210.7980",
  "version": "v1",
  "published": "2012-10-30T12:25:09.000Z",
  "updated": "2012-10-30T12:25:09.000Z",
  "title": "On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data",
  "authors": [
    "Bingbing Ding",
    "Ingo Witt",
    "Huicheng Yin"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\\p_t^2u-\\ds\\sum_{i=1}^2\\p_i(c_i^2(u)\\p_iu)$ $=0$, where $c_i(u)\\in C^{\\infty}(\\Bbb R^n)$, $c_i(0)\\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \\p_tu(0,x))=(\\ve u_0(x), \\ve u_1(x))$ with $u_0(x), u_1(x)\\in C_0^{\\infty}(\\Bbb R^2)$, and $\\ve>0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\\na_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\\ve}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-30T12:25:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35J70",
      "35R35"
    ],
    "keywords": [
      "nonlinear wave equation",
      "small initial data",
      "blowup mechanism",
      "smooth solutions",
      "nonlinear variational wave equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.7980D"
    }
  }
}