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

Abstract

This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, \begin{align*} u_t=\Delta (u+1)^m - \chi \nabla \cdot (u(u+1)^{\alpha - 1} \nabla v) + \lambda(|x|) u - \mu(|x|) u^\kappa, \quad 0=\Delta v - v + u, \quad x\in\Omega,\ t>0, \end{align*} where $\Omega:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$ is a ball with some $R>0$; $m>0$, $\chi>0$, $\alpha>0$ and $\kappa\ge1$; $\lambda$ and $\mu$ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for $\kappa$ such that solutions blow up in finite time when $m=\alpha=1$. In the case that $m=1$ and $\alpha\in(0,1)$ as well as $\lambda$ and $\mu$ are constant, some conditions for $\alpha$ and $\kappa$ 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 $\alpha=1$ 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 $\kappa$. The purpose of the present paper is to give conditions for $m\ge1$, $\alpha>0$ and $\kappa\ge1$ such that solutions blow up in finite time.