{
  "id": "1501.05175",
  "version": "v1",
  "published": "2015-01-21T14:17:00.000Z",
  "updated": "2015-01-21T14:17:00.000Z",
  "title": "A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity",
  "authors": [
    "Johannes Lankeit"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the parabolic chemotaxis model \\[ u_t=\\Delta u - \\chi \\nabla\\cdot(\\frac uv \\nabla v), \\qquad\\qquad v_t=\\Delta v - v + u\\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $\\chi\\in(0,\\chi_0)$ for some $\\chi_0>1$, thereby proving that the value $\\chi=1$ is not critical in this regard. Our main tool is consideration of the energy functional \\[ \\mathcal{F}_{a,b}(u,v)=\\int_\\Omega u\\ln u - a \\int_\\Omega u\\ln v + b \\int_\\Omega |\\nabla \\sqrt{v}|^2 \\] for $a>0$, $b\\geq 0$, where using nonzero values of $b$ appears to be new in this context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-21T14:17:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35A01",
      "35A09",
      "92C17",
      "35A07",
      "35B40"
    ],
    "keywords": [
      "two-dimensional parabolic chemotaxis system",
      "singular sensitivity",
      "boundedness",
      "parabolic chemotaxis model",
      "convex two-dimensional domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}