{
  "id": "1207.5306",
  "version": "v2",
  "published": "2012-07-23T06:39:51.000Z",
  "updated": "2012-07-30T03:06:58.000Z",
  "title": "Blow up for some semilinear wave equations in multi-space dimensions",
  "authors": [
    "Yi Zhou",
    "Wei Han"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AP",
    "math.CT"
  ],
  "abstract": "In this paper, we discuss a new nonlinear phenomenon. We find that in $n\\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \\label{0.1} \\Box u(t,x) = |u(t,x)|^{q}, \\ \\ x\\in R^{n}, {equation} and {equation} \\label{0.2} \\Box u(t,x) = |u_{t}(t,x)|^{p}, \\ \\ x\\in R^{n} {equation} both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation {equation} \\label{0.3} \\Box u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \\ \\ x\\in R^{n}, {equation} with small initial data will blow up in finite time. In the two dimensional case, we also find that if $ q=4$, the Cauchy problem for the equation \\eqref{0.1} has global existence, and the Cauchy problem for the equation {equation} \\label{0.4} \\Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \\ \\ x\\in R^{2} {equation} has almost global existence, that is, the life span is at least $ \\exp (c\\varepsilon^{-2}) $ for initial data of size $ \\varepsilon$. However, in the combined nonlinearity case, the Cauchy problem for the equation {equation} \\label{0.5} \\Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \\ \\ x\\in R^{2} {equation} has a life span which is of the order of $ \\varepsilon^{-18} $ for the initial data of size $ \\varepsilon$, this is considerably shorter in magnitude than that of the first two equations. This solves an open optimality problem for general theory of fully nonlinear wave equations (see \\cite{Katayama}).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-30T03:06:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear wave equations",
      "cauchy problem",
      "nonlinear wave equation",
      "multi-space dimensions",
      "small initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5306Z"
    }
  }
}