{
  "id": "1510.07173",
  "version": "v1",
  "published": "2015-10-24T18:32:08.000Z",
  "updated": "2015-10-24T18:32:08.000Z",
  "title": "Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant",
  "authors": [
    "Tobias Black"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \\begin{align*} \\left\\{\\begin{array}{c@{\\,}l@{\\quad}l@{\\,}c} u_{t}&=\\Delta u-\\nabla\\!\\cdot(u\\nabla v),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ 0 &=\\Delta v+u+f(x),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ u(x,0)&=u_{0}(x),\\ &x\\in\\mathbb{R}^n,& \\end{array}\\right. \\end{align*} with prototypical external signal production \\begin{align*} f(x):=\\begin{cases} f_0\\vert x\\vert^{-\\alpha},&\\text{ if }\\vert x\\vert \\leq R-\\rho,\\\\ 0,&\\text{ if } \\vert x\\vert\\geq R+\\rho,\\\\ \\end{cases} \\end{align*} for $R\\in(0,1)$ and $\\rho\\in\\left(0,\\frac{R}{2}\\right)$, which is still integrable but not of class $\\text{L}^{\\frac{n}{2}+\\delta_0}(\\mathbb{R}^n)$ for some $\\delta_0\\in[0,1)$. For corresponding parabolic-parabolic Neumann-type boundary-value problems in bounded domains $\\Omega$, where $f\\in\\text{L}^{\\frac{n}{2}+\\delta_0}(\\Omega)\\cap C^{\\alpha}(\\Omega)$ for some $\\delta_0\\in(0,1)$ and $\\alpha\\in(0,1)$, it is known that the system does not emit blow-up solutions if the quantities $\\|u_0\\|_{\\text{L}^{\\frac{n}{2}+\\delta_0}(\\Omega)}, \\|f\\|_{\\text{L}^{\\frac{n}{2}+\\delta_0}(\\Omega)}$ and $\\|v_0\\|_{\\text{L}^{\\theta}(\\Omega)}$, for some $\\theta>n$, are all bounded by some $\\varepsilon>0$ small enough. We will show that whenever $f_0>\\frac{2n}{\\alpha}(n-2)(n-\\alpha)$ and $u_0\\equiv c_0>0$ in $\\overline{B_1(0)}$, a measure-valued global-in-time weak solution to the system above can be constructed which blows up immediately. Since these conditions are independent of $R\\in(0,1)$ and $c_0>0$, we will thus prove the criticality of $\\delta_0=0$ for the existence of global bounded solutions under a smallness conditions as described above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-24T18:32:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K40",
      "35B44",
      "35K61",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "chemotaxis system",
      "external chemoattractant",
      "nonnnegative radially symmetric solutions",
      "corresponding parabolic-parabolic neumann-type boundary-value problems",
      "global-in-time weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007173B"
    }
  }
}