Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

Abstract: We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system \begin{align*} \begin{cases} u_t = \Delta u - \nabla\cdot(u\nabla v) + \kappa(x)u-\mu(x)u^2, \ v_t = \Delta v - v + u \end{cases} \end{align*} in bounded domains $\Omega\subset \mathbb{R}^2$ and show that the blow-up set is contained in the set of zeroes of $\mu$.