{
  "id": "1811.09007",
  "version": "v1",
  "published": "2018-11-22T03:39:43.000Z",
  "updated": "2018-11-22T03:39:43.000Z",
  "title": "Global Stability of Keller--Segel Systems in Critical Lebesgue Spaces",
  "authors": [
    "Jie Jiang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the global stability of classical solutions to a Keller--Segel equations in scaling-invariant spaces. We prove that for any given $0<\\mathcal{M}<1+\\lambda_1$ with $\\lambda_1$ being the first eigenvalue of Neumann Laplacian, the initial--boundary value problem of the Keller--Segel system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to $(\\mathcal{M},\\mathcal{M})$ in the norm of $L^{d/2}(\\Omega)\\times \\dot{W}^{1,d}(\\Omega)$ and satisfies a natral average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller--Segel equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-22T03:39:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical lebesgue spaces",
      "keller-segel system",
      "global stability",
      "globally bounded classical solution",
      "initial datum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}