Finite-time blow-up in a repulsive chemotaxis-consumption system

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$ , the chemotaxis system \[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \] is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$ . It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi >0$ with some ${K_D}>0$ and $\alpha >0$ , then for all initial data from a considerably large set of radial functions on $\Omega$ , the corresponding initial-boundary value problem admits a solution blowing up in finite time.