{
  "id": "1809.06155",
  "version": "v1",
  "published": "2018-09-17T12:14:59.000Z",
  "updated": "2018-09-17T12:14:59.000Z",
  "title": "Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up",
  "authors": [
    "Laurent Lafleche",
    "Samir Salem"
  ],
  "comment": "30 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article studies the aggregation diffusion equation \\[ \\partial_t\\rho = \\Delta^\\frac{\\alpha}{2} \\rho + \\lambda\\,\\mathrm{div}((K*\\rho)\\rho), \\] where $\\Delta^\\frac{\\alpha}{2}$ denotes the fractional Laplacian and $K = \\frac{x}{|x|^a}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case $a < \\alpha$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $a = \\alpha$ for an $L^1_k\\cap L\\ln L$ initial condition. In the aggregation dominated case $a > \\alpha$, we prove global or local well posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial condition. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-17T12:14:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "35A01",
      "35A02",
      "35B44",
      "35B40"
    ],
    "keywords": [
      "finite time blow-up",
      "fractional keller-segel equation",
      "global well-posedness",
      "initial condition",
      "initial mass concentration criteria"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}