{
  "id": "2205.08761",
  "version": "v1",
  "published": "2022-05-18T07:08:28.000Z",
  "updated": "2022-05-18T07:08:28.000Z",
  "title": "Keller-Segel model with Logarithmic Interaction and nonlocal reaction term",
  "authors": [
    "Shen Bian",
    "Quan Wang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term $u\\left(M_0-\\int_{\\R^2} u dx\\right)$ in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value $8\\pi$ for the growth parameter $M_0$ and the initial mass $m_0$. For general solutions, if both $m_0$ and $M_0$ are less than $8\\pi$, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for $M_0>8\\pi$ (It involves the case $m_0<8\\pi$) with any initial data and $M_0<8\\pi<m_0$ with small initial second moment. We also show the infinite time blow-up for the critical case $M_0=8 \\pi.$ Moreover, in the radial context, we show that if the initial data $u_0(r)<\\frac{m_0}{M_0} \\frac{8 \\lambda}{(r^2+\\lambda)^2}$ for some $\\lambda>0$, then all the radially symmetric solutions are vanishing in $L_{loc}^1(\\R^2)$ as $t \\to \\infty$. If the initial data $u_0(r)>\\frac{m_0}{M_0} \\frac{8 \\lambda}{(r^2+\\lambda)^2}$ for some $\\lambda>0$, then there could exist a radially symmetric solution satisfying a mass concentration at the origin as $t \\to \\infty.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T07:08:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal reaction term",
      "keller-segel model",
      "logarithmic interaction",
      "initial data",
      "radially symmetric solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}