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

This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, \begin{align*} u_t=Δ(u+1)^m - χ\nabla \cdot (u(u+1)^{α- 1} \nabla v) + λ(|x|) u - μ(|x|) u^κ, \quad 0=Δv - v + u, \quad x\inΩ,\ t>0, \end{align*} where $Ω:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$ is a ball with some $R>0$; $m>0$, $χ>0$, $α>0$ and $κ\ge1$; $λ$ and $μ$ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for $κ$ such that solutions blow up in finite time when $m=α=1$. In the case that $m=1$ and $α\in(0,1)$ as well as $λ$ and $μ$ are constant, some conditions for $α$ and $κ$ such that blow-up occurs were obtained in a previous paper (Math. Methods Appl. Sci.; 2020; 43; 7372-7396). Moreover, in the case that $m\ge1$ and $α=1$ Black, Fuest and Lankeit (arXiv:2005.12089[math.AP]) showed that there exists initial data such that solutions blow up in finite time under some conditions for $m$ and $κ$. The purpose of the present paper is to give conditions for $m\ge1$, $α>0$ and $κ\ge1$ such that solutions blow up in finite time.