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=\Delta(u+1)^m-\nabla\cdot(u\chi(v)\nabla v),\quad &x\in\Omega, t>0,\\ 0=\Delta v-v+u, &x\in\Omega, t>0 \end{cases} \end{align} under homogeneous Newmann boundary conditions and initial conditions, where $\Omega=B_R(0)\subset\mathbb{R}^N$ ($N\geq3,\ R>0$) is a ball, $m\geq 1$, $\chi$ is a function satisfying that $\chi(s)\geq\chi_0(a+s)^{-k}$ ($k>0$, $\chi_0>0$, $a\geq 0$) for all $s>0$ and some conditions. If the case that $m=1$ and $\chi(s)=\chi_0s^{-k}$, 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 $\chi(s)\equiv(\mbox{const.})$, 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.