{
  "id": "1705.06445",
  "version": "v1",
  "published": "2017-05-18T07:28:30.000Z",
  "updated": "2017-05-18T07:28:30.000Z",
  "title": "Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity",
  "authors": [
    "Tobias Black"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate the parabolic-elliptic Keller-Segel model \\{ {r@{\\,}l@{\\quad}l@{\\quad}l@{\\,}c} u_{t}&\\!\\!\\!=\\Delta u-\\,\\chi\\nabla\\!\\cdot(\\frac{u}{v}\\nabla v),\\ &x\\in\\Omega,& t>0, 0&\\!\\!\\!=\\Delta v-\\,v+u,\\ &x\\in\\Omega,& t>0, \\frac{\\partial u}{\\partial\\nu}&\\!\\!\\!=\\frac{\\partial v}{\\partial\\nu}=0,\\ &x\\in\\partial\\Omega,& t>0, u(&\\!\\!\\!\\!\\!\\!x,0)=u_0(x),\\ &x\\in\\Omega,& in a bounded domain $\\Omega\\subset\\mathbb{R}^n$ $(n\\geq2)$ with smooth boundary. We introduce a notion of generalized solvability which is consistent with the classical solution concept, and we show that whenever $0<\\chi<\\frac{n}{n-2}$ and the initial data satisfy only certain requirements on regularity and on positivity, one can find at least one global generalized solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-18T07:28:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35D99",
      "35A01",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "global generalized solution",
      "parabolic-elliptic keller-segel system",
      "singular sensitivity",
      "initial data satisfy",
      "parabolic-elliptic keller-segel model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}