{
  "id": "2406.11532",
  "version": "v1",
  "published": "2024-06-17T13:36:25.000Z",
  "updated": "2024-06-17T13:36:25.000Z",
  "title": "Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with general nonlinearity in the radial case",
  "authors": [
    "Jun Wang",
    "Zhaoyang Yin"
  ],
  "comment": "46. arXiv admin note: substantial text overlap with arXiv:math/0610266 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\\\"odinger equation with general nonlinearity \\begin{equation*} \\left\\{\\begin{array}{l} i \\partial_t u+\\Delta u + f(u)=0,\\ (x, t) \\in \\mathbb{R}^N \\times \\mathbb{R}, \\\\ \\left.u\\right|_{t=0}=u_0 \\in H ^1(\\mathbb{R}^N), \\end{array}\\right. \\end{equation*} where $f:\\mathbb{C}\\rightarrow \\mathbb{C}$ satisfies Sobolev critical growth condition. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. This paper generalizes the conclusions in \\cite{KCEMF2006}(Invent. Math).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-17T13:36:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general nonlinearity",
      "radial case",
      "schrödinger equation",
      "global well-posedness",
      "well-posedness theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}