{
  "id": "math/0110321",
  "version": "v6",
  "published": "2001-10-31T01:30:01.000Z",
  "updated": "2003-11-10T17:42:04.000Z",
  "title": "Almost global existence for quasilinear wave equations in three space dimensions",
  "authors": [
    "M. Keel",
    "H. Smith",
    "C. D. Sogge"
  ],
  "comment": "This revised version of our paper will appear in the Journal of the American Mathematical Society",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equations that gives $O(1/t)$ decay of solutions of the inhomomogeneous linear wave equation based in terms of $O(1/|x|)$ estimates for the forcing term.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2003-11-10T17:42:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L70",
      "42B99"
    ],
    "keywords": [
      "global existence",
      "space dimensions",
      "nonlinear dirichlet-wave equations outside",
      "minkowski space",
      "multiple speed quasilinear wave equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....10321K"
    }
  }
}