{
  "id": "2103.02246",
  "version": "v1",
  "published": "2021-03-03T08:15:46.000Z",
  "updated": "2021-03-03T08:15:46.000Z",
  "title": "Boundedness in a fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion and singular sensitivity",
  "authors": [
    "Yutaro Chiyo",
    "Tomomi Yokota"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \\begin{align*} u_t=\\nabla \\cdot (D(u)\\nabla u) -\\nabla \\cdot (G(u)\\chi(v)\\nabla v) +\\nabla\\cdot(H(u)\\xi(w)\\nabla w), \\quad v_t=d_1\\Delta v+\\alpha u-\\beta v, \\quad w_t=d_2\\Delta w+\\gamma u-\\delta w, \\quad x \\in \\Omega,\\ t>0, \\end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\\Omega \\subset \\mathbb{R}^n$ $(n \\ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, \\alpha, \\beta, \\gamma, \\delta>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \\in \\mathbb{R}$; $0 \\le G(s) \\le b_0(s+1)^{q-1}$ with $b_0>0$ and $q<\\min\\{2,\\ m+1\\}$; $0 \\le H(s) \\le c_0(s+1)^{r-1}$ with $c_0>0$ and $r<\\min\\{2,\\ m+1\\}$, and the signal-dependent sensitivities $\\chi, \\xi$ satisfy $0<\\chi(s)\\le \\frac{\\chi_0}{s^{k_1}}$ with $\\chi_0>0$ and $k_1>1$; $0<\\xi(s)\\le \\frac{\\xi_0}{s^{k_2}}$ with $\\xi_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (J. Math. Anal. Appl.; 2018;461;1260-1270) and Jia-Yang (J. Math. Anal. Appl.; 2019;475;139-153). However, there has been no work on the above fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion and singular sensitivity. This paper develops global existence and boundedness of classical solutions to the above system by introducing a new test function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-03T08:15:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "fully parabolic attraction-repulsion chemotaxis system",
      "nonlinear diffusion",
      "singular sensitivity",
      "boundedness",
      "quasilinear fully parabolic attraction-repulsion chemotaxis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}