{
  "id": "2406.12633",
  "version": "v1",
  "published": "2024-06-18T14:01:34.000Z",
  "updated": "2024-06-18T14:01:34.000Z",
  "title": "Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity",
  "authors": [
    "Gregor Flüchter"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \\begin{eqnarray*} \\left\\{ \\begin{array}{l} u_t = \\nabla \\cdot (|x|^\\beta \\nabla u) - \\nabla \\cdot (u\\nabla v), \\\\[1mm] 0 = \\Delta v - \\mu + u, \\qquad \\mu:=\\frac{1}{|\\Omega|} \\int\\limits_\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star) \\end{eqnarray*} under homogeneous Neumann boundary conditions in the ball $\\Omega=B_R(0)\\subset \\mathbb R^n$. For $\\beta>0$ and radially symmetric H\\\"older continuous initial data, we prove that there exists a pointwise classical solution to $(\\star)$ in $(\\Omega\\setminus \\{0\\})\\times (0,T)$ for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in $(\\Omega\\setminus \\{0\\})\\times (0,T^*)$ for some $T^*>0$. Moreover, for $n \\geq 2$ and sufficiently accumulated initial data, there exists no solution $(u,v)$ to $(\\star)$ in the sense specified above which is globally bounded in time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-18T14:01:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35K65",
      "35A02",
      "35B44",
      "35B33",
      "92C17"
    ],
    "keywords": [
      "spatially heterogeneous diffusion sensitivity",
      "chemotaxis system",
      "decreasing initial data satisfying",
      "homogeneous neumann boundary conditions",
      "parabolic-elliptic keller-segel system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}