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 $$\begin{array}{rl}{\displaystyle u_t}& {\displaystyle =\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (\frac{u}{v}\mathrm{\nabla }v)+\kappa u-\mu u^2}\\ {\displaystyle v_t}& {\displaystyle =\mathrm{\Delta }v-uv}\end{array}$$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, 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 $\chi >0$, $\mu >0$, $\kappa \ge 0$.