{
  "id": "2201.08716",
  "version": "v1",
  "published": "2022-01-21T14:27:11.000Z",
  "updated": "2022-01-21T14:27:11.000Z",
  "title": "Blow-up phenomena for a chemotaxis system with flux limitation",
  "authors": [
    "M. Marras",
    "S. Vernier-Piro",
    "T. Yokota"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \\begin{equation*} \\left\\{ \\begin{array}{l} \\begin{aligned} &u_t = \\Delta u - \\nabla(u f(|\\nabla v|^2 )\\nabla v), \\\\[6pt] &0= \\Delta v -\\mu + u , \\quad \\int_{\\Omega}v =0, \\ \\ \\mu := \\frac 1 {|\\Omega|} \\int_{\\Omega} u dx, \\\\[6pt] &u(x,0)= u_0(x), \\end{aligned} \\end{array} \\right. \\end{equation*} in $\\Omega \\times (0,\\infty)$, with $\\Omega$ a ball in $\\mathbb{R}^N$, $N\\geq 3$ under homogeneous Neumann boundary conditions and $f(\\xi) = (1+ \\xi)^{-\\alpha}$, $0<\\alpha < \\frac{N-2}{2(N-1)}$, which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on $f$ and initial data, we prove that a solution which blows up in finite time in $L^\\infty$-norm, blows up also in $L^p$-norm for some $p>1$. Moreover, a lower bound of blow-up time is derived. \\vskip.2truecm \\noindent{\\bf AMS Subject Classification }{Primary: 35B44; Secondary: 35Q92, 92C17.} \\vskip.2truecm \\noindent{\\bf Key Words:} finite-time blow-up; chemotaxis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-21T14:27:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B44",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "blow-up phenomena",
      "chemotaxis system",
      "flux limitation",
      "ams subject classification",
      "parabolic-elliptic cross-diffusion system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}