Finite time blow-up in a parabolic-elliptic Keller-Segel system with nonlinear diffusion and signal-dependent sensitivity

Abstract

This paper is concerned with the parabolic-elliptic Keller-Segel system with nonlinear diffusion and signal-dependent sensitivity \begin{align}\tag{KS}\label{system} \begin{cases} u_t=Δ(u+1)^m-\nabla\cdot(uχ(v)\nabla v),\quad &x\inΩ, t>0,\\ 0=Δv-v+u, &x\inΩ, t>0 \end{cases} \end{align} under homogeneous Newmann boundary conditions and initial conditions, where $\Omega =B_R(0)\subset {\mathbb{R}}^N$ ($N\ge 3,R>0$) is a ball, $m\ge 1$, $\chi$ is a function satisfying that $\chi (s)\ge {\chi }_0(a+s{)}^{-k}$ ($k>0$, ${\chi }_0>0$, $a\ge 0$) for all $s>0$ and some conditions. If the case that $m=1$ and $\chi (s)={\chi }_0{s}^{-k}$, Nagai-Senba established finite-time blow-up of solutions under the smallness conditions on a moment of initial data $u(x,0)$ and some condition for $k\in (0,1)$. Moreover, if the case that $χ(s)\equiv(\mbox{const.})$, Sugiyama showed finite-time blow-up of solutions under the condition $m\in [1,2-\frac{2}{N})$. According to two previous works, it seems that the smallness conditions of $m$ and $k$ leads to finite-time blow-up of solutions. The purpose of this paper is to give the relationship which depends only on $m$, $k$ and $N$ such that there exists initial data which corresponds finite-time blow-up solutions.

References in corpus (1)

• Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source