Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption

Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing $$\begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v} \nabla v ) + \kappa u -\mu u^2\\ v_t &= \Delta v - u v \end{split}$$ in a bounded domain $\Omega \subset \mathbb{R}^n$, with homogeneous Neumann boundary conditions and for widely arbitrary positive initial data. In the spatially one-dimensional setting, we prove global existence and, moreover, boundedness of the solution for any $\chi>0$, $\mu>0$, $\kappa\ge 0$.