{
  "id": "1402.3879",
  "version": "v1",
  "published": "2014-02-17T03:26:24.000Z",
  "updated": "2014-02-17T03:26:24.000Z",
  "title": "A Semi-linear Shifted Wave Equation on the Hyperbolic Spaces with Application on a Quintic Wave Equation on ${\\mathbb R}^2$",
  "authors": [
    "Ruipeng Shen",
    "Gigliola Staffilani"
  ],
  "comment": "51 pages, 9 figures, 4 tables",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space \\[ \\partial_t^2 u - (\\Delta_{{\\mathbb H}^n} + \\rho^2) u = - |u|^{p-1} u, \\quad (x,t)\\in {\\mathbb H}^n \\times {\\mathbb R}; \\] and introduce a Morawetz-type inequality \\[ \\int_{-T_-}^{T_+} \\int_{{\\mathbb H}^n} |u|^{p+1} d\\mu dt < C E, \\] where $E$ is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in $H^{1/2,1/2} \\times H^{1/2,-1/2}({\\mathbb H}^n)$ if $2 \\leq n \\leq 6$ and $1<p<p_c = 1+ 4/(n-2)$. As another application we show that a solution to the quintic wave equation $\\partial_t^2 u - \\Delta u = - |u|^4 u$ on ${\\mathbb R}^2$ scatters if its initial data are radial and satisfy the conditions \\[ |\\nabla u_0 (x)|, |u_1 (x)| \\leq A(|x|+1)^{-3/2-\\varepsilon};\\quad |u_0 (x)| \\leq A(|x|)^{-1/2-\\varepsilon};\\quad \\varepsilon >0. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-17T03:26:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L71",
      "35L05"
    ],
    "keywords": [
      "semi-linear shifted wave equation",
      "quintic wave equation",
      "hyperbolic space",
      "application",
      "morawetz-type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3879S"
    }
  }
}