{
  "id": "1701.02817",
  "version": "v1",
  "published": "2017-01-11T01:01:26.000Z",
  "updated": "2017-01-11T01:01:26.000Z",
  "title": "A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity",
  "authors": [
    "Masaaki Mizukami",
    "Tomomi Yokota"
  ],
  "comment": "15pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with the Keller--Segel system with signal-dependent sensitivity \\begin{equation*} u_t=\\Delta u - \\nabla \\cdot (u \\chi(v)\\nabla v), \\quad v_t=\\Delta v + u - v, \\quad x\\in\\Omega,\\ t>0, \\end{equation*} where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$; $\\chi$ is a function satisfying $\\chi(s)\\leq K(a+s)^{-k}$ for some $k\\geq 1$ and $a\\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\\sqrt{\\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted global existence of bounded solutions for arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for $\\chi$ and to build a mathematical bridge between the cases $k=1$ and $k>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-11T01:01:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K51",
      "35B45",
      "35A01",
      "92C17"
    ],
    "keywords": [
      "fully parabolic chemotaxis systems",
      "signal-dependent sensitivity",
      "unified method",
      "bounded solutions",
      "boundedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}