{
  "id": "2409.14890",
  "version": "v1",
  "published": "2024-09-23T10:36:52.000Z",
  "updated": "2024-09-23T10:36:52.000Z",
  "title": "Absence of dead-core formations in chemotaxis systems with degenerate diffusion",
  "authors": [
    "Tobias Black"
  ],
  "comment": "7 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider a chemotaxis system with signal consumption and degenerate diffusion of the form \\begin{align*} \\left\\lbrace \\begin{array}{r@{}l@{\\quad}l} &u_t=\\nabla\\cdot\\big(D(u)\\nabla u-uS(u)\\nabla v\\big)+f(u,v),\\\\ &v_t=\\Delta v- uv,\\\\ \\end{array}\\right. \\end{align*} in a bounded domain $\\Omega\\subset\\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\\in C^0([0,\\infty))\\cap C^2((0,\\infty))$ is assumed to satisfy $D(0)=0$, $D(s)>0$ on $(0,\\infty)$, $D'(s)\\geq 0$ on $(0,\\infty)$ and that there are $s_0>0$, $p>1$ and $C_D>0$ such that $$s D'(s)\\leq C_D D(s)\\quad\\text{and}\\quad C_D s^{p-1}\\leq D(s)\\quad\\text{for }s\\in[0,s_0].$$ The sensitivity function $S\\in C^2([0,\\infty))$ and the source term $f\\in C^{1}([0,\\infty)\\times[0,\\infty))$ are supposed to be nonnegative. We show that for all suitably regular initial data $(u_0,v_0)$ satisfying $u_0\\geq \\delta_0>0$ and $v_0\\not\\equiv 0$ there is a time-local classical solution and - despite the degeneracy at $0$ - the solution satisfies an extensibility criterion of the form $$\\text{either}\\quad T_{max}=\\infty,\\quad\\text{or}\\quad\\limsup_{t\\nearrow T_{max}}\\|u(\\cdot,t)\\|_{L^\\infty(\\Omega)}=\\infty.$$ Moreover, as a by-product of our analysis, we prove that a classical solution on $\\Omega\\times(0,T)$ obeying $\\|u(\\cdot,t)\\|_{L^\\infty(\\Omega)}\\leq M_u$ for all $t\\in(0,T)$ and emanating from initial data $(u_0,v_0)$ as specified above remains strictly positive throughout $\\Omega\\times(0,T)$, i.e. one can find $\\delta_u=\\delta_u(T,\\delta_0, M_u,\\|v_0\\|_{W^{1,\\infty}(\\Omega)})>0$ such that $$u(x,t)\\geq\\delta_u\\quad\\text{for all }(x,t)\\in\\Omega\\times(0,T).$$ Together, the results indicate that the formation of a dead-core in these chemotaxis systems with a degenerate diffusion are impossible before the blow-up time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-23T10:36:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35B09",
      "35K65",
      "35Q92",
      "92C17"
    ],
    "keywords": [
      "chemotaxis system",
      "degenerate diffusion",
      "dead-core formations",
      "homogeneous neumann boundary conditions",
      "suitably regular initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}