On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms

Introducing a suitable solution concept, we show that in bounded smooth domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=\Delta u -\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)+\kappa u -\mu u^2,\ v_t&=\Delta v -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $\mu ,\kappa ,\chi >0$.