Finite-time blow-up in a two-dimensional Keller--Segel system with an environmental dependent logistic source

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa(|x|) u - \mu(|x|) u^p, \\ 0 = \Delta v - \frac{m(t)}{|\Omega|} + u, \quad m(t) := \int_\Omega u(\cdot, t) \end{cases} \end{align} is studied in a ball $\Omega = B_R(0) \subset \mathbb R^2$, $R \gt 0$ for $p \ge 1$ and sufficiently smooth functions $\kappa, \mu: [0, R] \rightarrow [0, \infty)$. We prove that whenever $\mu', -\kappa' \ge 0$ as well as $\mu(s) \le \mu_1 s^{2p-2}$ for all $s \in [0, R]$ and some $\mu_1 \gt 0$ then for all $m_0 \gt 8 \pi$ there exists $u_0 \in C^0(\overline \Omega)$ with $\int_\Omega u_0 = m_0$ and a solution $(u, v)$ to \eqref{prob:abstract} with initial datum $u_0$ blowing up in finite time. If in addition $\kappa \equiv 0$ then all solutions with initial mass smaller than $8 \pi$ are global in time, displaying a certain critical mass phenomenon. On the other hand, if $p \gt 2$, we show that for all $\mu$ satisfying $\mu(s) \ge \mu_1 s^{p-2-\varepsilon}$ for all $s \in [0, R]$ and some $\mu_1, \varepsilon \gt 0$ the system \eqref{prob:abstract} admits a global classical solution for each initial datum $0 \le u_0 \in C^0(\overline \Omega)$