{
  "id": "1112.4618",
  "version": "v1",
  "published": "2011-12-20T09:31:54.000Z",
  "updated": "2011-12-20T09:31:54.000Z",
  "title": "The dynamics of the NLS with the combined terms in five and higher dimensions",
  "authors": [
    "Changxing Miao",
    "Guixiang Xu",
    "Lifeng Zhao"
  ],
  "comment": "33pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we continue the study in \\cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schr\\\"{o}dinger equation with the energy below the threshold $m$ in the energy space $H^1(\\R^d)$, iu_t + \\Delta u = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \\; d\\geq 5. \\tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \\Delta u = -|u|^{4/(d-2)}u$. Compared with the argument in \\cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \\cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L^{\\infty}_tH^1_x$ to $L^{\\infty}\\dot H^{-\\epsilon}_x$ for some $\\epsilon>0$ in five and higher dimensions and obtain the compactness of the critical element in $L^2_x$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-20T09:31:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher dimensions",
      "critical element",
      "spatial center function",
      "ground state",
      "blow-up result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.4618M"
    }
  }
}