{
  "id": "1103.2530",
  "version": "v3",
  "published": "2011-03-13T17:11:09.000Z",
  "updated": "2011-08-03T16:30:17.000Z",
  "title": "On the meromorphic continuation of the resolvent for the wave equation with time-periodic perturbation and applications",
  "authors": [
    "Yavar Kian"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Consider the wave equation $\\partial_t^2u-\\Delta_xu+V(t,x)u=0$, where $x\\in\\R^n$ with $n\\geq3$ and $V(t,x)$ is $T$-periodic in time and decays exponentially in space. Let $ U(t,0)$ be the associated propagator and let $R(\\theta)=e^{-D<x>}(U(T,0)-e^{-i\\theta})^{-1}e^{-D<x>}$ be the resolvent of the Floquet operator $U(T,0)$ defined for $\\im(\\theta)>BT $ with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(\\theta)$ from which we deduce the asymptotic expansion of $e^{-(D+\\epsilon)<x>}U(t,0)e^{-D<x>}f$, where $f\\in \\dot{H}^1(\\R^n)\\times L^2(\\R^n)$, as $t\\to+\\infty$ with a remainder term whose energy decays exponentially when $n$ is odd and a remainder term whose energy is bounded with respect to $t^l\\log(t)^m$, with $l,m\\in\\mathbb Z$, when $n$ is even. Then, assuming that $R(\\theta)$ has no poles lying in $\\{\\theta\\in\\C\\ :\\ \\im(\\theta)\\geq0\\}$ and is bounded for $\\theta\\to0$, we obtain local energy decay as well as global Strichartz estimates for the solutions of $\\partial_t^2u-\\Delta_xu+V(t,x)u=F(t,x)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-08-03T16:30:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "meromorphic continuation",
      "wave equation",
      "time-periodic perturbation",
      "remainder term",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.2530K"
    }
  }
}