Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source

We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth systems of the form \begin{align*} u_t &= Δu - \nabla\cdot(u\nabla v) + κ(x)u-μ(x)u^2, 0 &= Δv - v + u \end{align*} in smoothly bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$. Assuming that the coefficient functions satisfy $\kappa ,\mu \in C^0(\bar{\mathrm{\Omega }})$ with $\mu \ge 0$ we prove that finite-time blow-up of the classical solution can only occur in points where $\mu$ is zero, i.e.\ that the blow-up set $\mathcal{B}$ is contained in \begin{align*} \big{x\in\overlineΩ\midμ(x)=0\big}. \end{align*} Moreover, we show that whenever $\mu (x_0)>0$ for some $x_0\in \bar{\mathrm{\Omega }}$, then one can find an open neighbourhood $U$ of $x_0$ in $\bar{\mathrm{\Omega }}$ such that $u$ remains bounded in $U$ throughout evolution.