{
  "id": "1210.3429",
  "version": "v1",
  "published": "2012-10-12T04:36:48.000Z",
  "updated": "2012-10-12T04:36:48.000Z",
  "title": "Global Well-posedness of the Parabolic-parabolic Keller-Segel Model in $L^{1}(R^2)\\times{L}^{\\infty}(R^2)$ and $H^1_b(R^2)\\times{H}^1(R^2)$",
  "authors": [
    "Chao Deng",
    "Congming Li"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study global well-posedness of the two-dimensional Keller-Segel model in Lebesgue space and Sobolev space. Recall that in the paper \"Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, {252}(2012), 1213--1228\", Kozono, Sugiyama & Wachi studied global well-posedness of $n$($\\ge3$) dimensional Keller-Segel system and posted a question about the even local in time existence for the Keller-Segel system with $L^1(R^2)\\times{L}^\\infty(R^2)$ initial data. Here we give an affirmative answer to this question: in fact, we show the global in time existence and uniqueness for $L^1(R^2)\\times{L}^{\\infty}(R^2)$ initial data. Furthermore, we prove that for any $H^1_b(R^2) \\times {H}^1(R^2)$ initial data with $H^1_b(R^2):=H^1(R^2)\\cap{L}^\\infty(R^2)$, there also exists a unique global mild solution to the parabolic-parabolic Keller-Segel model. The estimates of ${\\sup_{t>0}}t^{1-\\frac{n}{p}}\\|u\\|_{L^p}$ for $(n,p)=(2,\\infty)$ and the introduced special half norm, i.e. $\\sup_{t>0}t^{1/2}(1+t)^{-1/2}\\|\\nabla{v}\\|_{L^\\infty}$, are crucial in our proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-12T04:36:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic-parabolic keller-segel model",
      "initial data",
      "unique global mild solution",
      "time existence",
      "two-dimensional keller-segel model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3429D"
    }
  }
}