{
  "id": "1902.09787",
  "version": "v1",
  "published": "2019-02-26T08:12:49.000Z",
  "updated": "2019-02-26T08:12:49.000Z",
  "title": "Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system",
  "authors": [
    "Teruto Nishino",
    "Tomomi Yokota"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \\begin{equation*} \\begin{cases} u_t=\\nabla \\cdot [(u+\\alpha)^{m_1-1} \\nabla u-\\chi u(u+\\alpha)^{m_2-2} \\nabla v] & {\\rm in} \\; \\Omega \\times (0,T), \\\\[1mm] v_t=\\Delta v-v+u & {\\rm in} \\; \\Omega \\times (0,T) \\end{cases} \\end{equation*} under Neumann boundary conditions and initial conditions, where $\\Omega$ is a general bounded domain in $\\mathbb{R}^n$ with smooth boundary, $\\alpha>0$, $\\chi>0$, $m_1, m_2 \\in \\mathbb{R}$ and $T>0$. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that $m_1=1$ and $\\Omega$ is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that $m_1 \\neq 1$ and $\\Omega$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $\\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T08:12:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fully parabolic chemotaxis system",
      "blow-up time",
      "lower bound",
      "nonlinear diffusion",
      "neumann boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}