{
  "id": "1703.08389",
  "version": "v1",
  "published": "2017-03-24T12:42:21.000Z",
  "updated": "2017-03-24T12:42:21.000Z",
  "title": "Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type",
  "authors": [
    "Masaaki Mizukami"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with the two-species chemotaxis-competition system $u_t = d_1 \\Delta u - \\chi_1 \\nabla \\cdot (u \\nabla w) + \\mu_1 u(1 - u - a_1 v)$, $v_t = d_2 \\Delta v - \\chi_2 \\nabla \\cdot (v \\nabla w) + \\mu_2 v(1 - a_2 u - v)$, $0 = d_3 \\Delta w + \\alpha u + \\beta v - \\gamma w$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$ with smooth boundary, $n\\ge 2$; $\\chi_i$ and $\\mu_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\\frac{\\chi_i}{\\mu_i}$ are small. However, the conditions in the above two cases strongly depend on $a_1,a_2$, and have not been obtained in the case that $a_1,a_2\\ge 1$. Moreover, convergence rates in the cases that $a_1,a_2\\in (0,1)$ and $a_1 > 1 > a_2$ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_1,a_2>0$ which covers the case that $a_1,a_2 \\ge 1$, and lead to convergence rates for solutions of the above system in the cases that $a_1,a_2\\in (0,1)$ and $a_1\\ge 1 >a_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-24T12:42:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K51",
      "92C17",
      "35B40"
    ],
    "keywords": [
      "two-species chemotaxis-competition system",
      "parabolic-parabolic-elliptic type",
      "global existence",
      "convergence rates",
      "boundedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}