{
  "id": "1509.08845",
  "version": "v1",
  "published": "2015-09-29T16:53:53.000Z",
  "updated": "2015-09-29T16:53:53.000Z",
  "title": "Blowup for fractional NLS",
  "authors": [
    "Thomas Boulenger",
    "Dominik Himmelsbach",
    "Enno Lenzmann"
  ],
  "comment": "25 pages. Comments are welcome",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider fractional NLS with focusing power-type nonlinearity $$i \\partial_t u = (-\\Delta)^s u - |u|^{2 \\sigma} u, \\quad (t,x) \\in \\mathbb{R} \\times \\mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < \\sigma < \\infty$ for $s \\geq N/2$ and $0 < \\sigma \\leq 2s/(N-2s)$ for $s < N/2$. We prove a general criterion for blowup of radial solutions in $\\mathbb{R}^N$ with $N \\geq 2$ for $L^2$-supercritical and $L^2$-critical powers $\\sigma \\geq 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $\\Omega \\subset \\mathbb{R}^N$ in any dimension $N \\geq 1$ and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on $u(t,x)$. For the blowup proof in $\\mathbb{R}^N$, we derive a localized virial estimate for fractional NLS in $\\mathbb{R}^N$, which uses Balakrishnan's formula for the fractional Laplacian $(-\\Delta)^s$ from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-29T16:53:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional nls",
      "fractional laplacian",
      "general blowup result",
      "exterior dirichlet conditions",
      "pohozaev-type estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}