Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production

This paper deals with finite-time blow-up of solutions to the quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production,\begin{document}$ \begin{align*} \begin{cases}u_t = \Delta u^m - \chi \nabla \cdot (u^\alpha \nabla v)+ \lambda u - \mu u^\kappa, \quad &x \in \Omega, \ t>0, \\0 = \Delta v - \overline{M_\ell}(t) + u^\ell, \quad &x \in \Omega, \ t>0, \end{cases} \end{align*} $\end{document}where \begin{document}$ \Omega: = B_R(0) \subset \mathbb{R}^n \ (n \in \mathbb{N}) $\end{document} be a ball with some \begin{document}$ R>0 $\end{document} and \begin{document}$ m\ge1 $\end{document} , \begin{document}$ \chi>0 $\end{document} , \begin{document}$ \alpha\ge1 $\end{document} , \begin{document}$ \lambda>0 $\end{document} , \begin{document}$ \mu>0 $\end{document} , \begin{document}$ \kappa>1 $\end{document} , \begin{document}$ \ell>0 $\end{document} as well as \begin{document}$ \overline{M_\ell}(t) $\end{document} is the average of \begin{document}$ u^\ell $\end{document} over \begin{document}$ \Omega $\end{document} . As to the corresponding system with nondegenerate diffusion, finite-time blow-up has been obtained under the condition that \begin{document}$ \alpha-\ell>\max\left\{\overline{m} +\frac{2}{n}\kappa, \kappa\right\} $\end{document} , where \begin{document}$ \overline{m}: = \max\{m,0\} $\end{document} in a previous paper [ 26 ], which is based a work by Fuest [ 7 ]. The purpose of this paper is to establish finite-time blow-up for the above degenerate chemotaxis system within a concept of weak solutions with a moment inequality leading to blow-up.