Finite-time blow-up in a quasilinear chemotaxis system with an external signal consumption

Abstract

This paper deals with a quasilinear chemotaxis system with an external signal consumption \begin{equation*} \begin{cases} u_t=\nabla\cdot(\varphi(u)\nabla u)-\nabla\cdot(u\nabla v), & (x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v+u-g(x), &(x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a ball $\Omega\subset \mathbb{R}^{n}$, where $\varphi(u)$ is a nonlinear diffusion function and $g(x)$ is an external signal consumption. Under suitable assumptions on the functions $\varphi$ and $g$, it is proved that there exists initial data such that the solution of the above system blows up in finite time.