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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 achemotaxis system slightly generalizing \[ \begin{split} u_t &= \Delta u - \chi\nabla\cdot ( \frac{u}{v} \nabla v ) + \kappa u -\mu u^2\\ v_t &= \Delta v - uv \end{split} \] in a bounded domain $\Omega \subset \mathbb{R}^n$, withhomogeneous Neumann boundary conditions and for widely arbitrary positiveinitial data. In the spatially one-dimensional setting, we prove globalexistence and, moreover, boundedness of the solution for any $\chi>0$, $\mu>0$,$\kappa\ge 0$.

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