{
  "id": "2210.07060",
  "version": "v1",
  "published": "2022-10-13T14:36:48.000Z",
  "updated": "2022-10-13T14:36:48.000Z",
  "title": "Some well-posedness and ill-posedness results for the INLS equation",
  "authors": [
    "Luccas Campos",
    "Simão Correia",
    "Luiz Gustavo Farah"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the initial value problem associated to the inhomogeneous nonlinear Schr\\\"o\\-din\\-ger equation, \\begin{equation} iu_t + \\Delta u +\\mu |x|^{-b}|u|^{\\alpha}u=0, \\quad u_0\\in H^s(\\mathbb R^N) \\text{ or } u_0 \\in\\dot H ^s(\\mathbb R^N), \\end{equation} with $\\mu=\\pm 1$, $b > 0$, $s\\geq 0$ and $0 < \\alpha \\leq \\frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove some ill-posedness results for this equation, through a delicate analysis of the associated Duhamel operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-13T14:36:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ill-posedness results",
      "inls equation",
      "initial value problem",
      "fractional leibniz rule",
      "local well-posedness results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}