{
  "id": "1903.08196",
  "version": "v1",
  "published": "2019-03-19T18:15:50.000Z",
  "updated": "2019-03-19T18:15:50.000Z",
  "title": "Explicit lower bound of blow--up time for an attraction--repulsion chemotaxis system",
  "authors": [
    "Giuseppe Viglialoro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \\begin{equation}\\label{ProblemAbstract} \\tag{$\\Diamond$} \\begin{cases} u_{ t}=\\Delta u -\\chi \\nabla \\cdot (u\\nabla v)+\\xi \\nabla \\cdot (u\\nabla w) & \\textrm{in }\\Omega\\times (0,t^*), \\\\ 0=\\Delta v+\\alpha u-\\beta v & \\textrm{in } \\Omega\\times (0,t^*),\\\\ 0=\\Delta w+\\gamma u-\\delta w & \\textrm{in } \\Omega\\times (0,t^*),\\\\ \\end{cases} \\end{equation} where $\\Omega$ is a smooth and bounded domain of $\\mathbb{R}^2$, $t^*$ is the blow--up time and $\\alpha,\\beta,\\gamma,\\delta,\\chi,\\xi$ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable smallness assumptions on the initial data $u({\\bf x},0)=u_0\\in C^0(\\bar{\\Omega})$, system \\eqref{ProblemAbstract} has a unique classical solution which becomes unbounded as $t\\nearrow t^*$. The main result of this investigation is to provide an explicit lower bound for $t^*$ estimated in terms of $\\int_\\Omega u_0^2 d{\\bf x}$ and attained by means of well--established techniques based on ordinary differential inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-19T18:15:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "explicit lower bound",
      "attraction-repulsion chemotaxis system",
      "blow-up time",
      "zero-flux attraction-repulsion chemotaxis-system",
      "ordinary differential inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}