{
  "id": "1904.03856",
  "version": "v1",
  "published": "2019-04-08T06:09:00.000Z",
  "updated": "2019-04-08T06:09:00.000Z",
  "title": "A refined criterion and lower bounds for the blow--up time in a parabolic--elliptic chemotaxis system with nonlinear diffusion",
  "authors": [
    "Monica Marras",
    "Teruto Nishino",
    "Giuseppe Viglialoro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with unbounded solutions to the following zero--flux chemotaxis system \\begin{equation}\\label{ProblemAbstract} \\tag{$\\Diamond$} \\begin{cases} % about u u_t=\\nabla \\cdot [(u+\\alpha)^{m_1-1} \\nabla u-\\chi u(u+\\alpha)^{m_2-2} \\nabla v] & (x,t) \\in \\Omega \\times (0,T_{max}), \\\\[1mm] % about v 0=\\Delta v-M+u & (x,t) \\in \\Omega \\times (0,T_{max}), \\end{cases} \\end{equation} where $\\alpha>0$, $\\Omega$ is a smooth and bounded domain of $\\mathbb{R}^n$, with $n\\geq 1$, $t\\in (0, T_{max})$, where $T_{max}$ the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth initial data $u_0:=u(x,0)\\geq 0$ and set $M:=\\frac{1}{|\\Omega|}\\int_{\\Omega}u_0(x)\\,dx$, from the literature it is known that under a proper interplay between the above parameters $m_1,m_2$ and the extra condition $\\int_\\Omega v(x,t)dx=0$, system \\eqref{ProblemAbstract} possesses for any $\\chi>0$ a unique classical solution which becomes unbounded at $t\\nearrow T_{max}$. In this investigation we first show that for $p_0>\\frac{n}{2}(m_2-m_1)$ any blowing up classical solution in $L^\\infty(\\Omega)$--norm blows up also in $L^{p_0}(\\Omega)$--norm. Then we estimate the blow--up time $T_{max}$ providing a lower bound $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-08T06:09:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "blow-up time",
      "parabolic-elliptic chemotaxis system",
      "lower bound",
      "nonlinear diffusion",
      "refined criterion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}