Finite time blow-up in a parabolic-elliptic Keller-Segel system with nonlinear diffusion and signal-dependent sensitivity
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 $Ω=B_R(0)\subset\mathbb{R}^N$ ($N\geq3,\ R>0$) is a ball, $m\geq 1$, $χ$ is a function satisfying that $χ(s)\geqχ_0(a+s)^{-k}$ ($k>0$, $χ_0>0$, $a\geq 0$) for all $s>0$ and some conditions. If the case that $m=1$ and $χ(s)=χ_0s^{-k}$, 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.