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$p \ge 1$andsufficientlysmoothfunctions$\kappa, \mu: [0, R] \rightarrow [0, \infty)$. We prove that whenever ${\mu }^{\mathrm{\prime }},-{\kappa }^{\mathrm{\prime }}\ge 0$ as well as $\mu(s) \le \mu_1 s^{2p-2}$forall$s \in [0, R]$andsome$\mu_1 \gt 0$thenforall$m_0 \gt 8 \pi$thereexists$u_0 \in C^0(\overline \Omega)$with$\int_\Omega u_0 = m_0$andasolution$(u, v)$to\text{eqref}prob:abstractwithinitialdatum$u_0$blowingupinfinitetime\mathrm{.}Ifinaddition$\kappa \equiv 0$ 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 $p>2$, we show that for all $\mu$ satisfying $\mu (s)\ge {\mu }_1{s}^{p-2-\epsilon }$ 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 $0\le u_0\in C^0(\bar{\mathrm{\Omega }})$