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In this paper we study the zero-flux chemotaxis-system \begin{equation*}\begin{cases} u_t=\Delta u -\chi \nabla \cdot (\frac{u}{v} \nabla v) \\v_t=\Delta v-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain$\Omega$ of $\mathbb{R}^2$, with $\chi>0$ and $f\in C^1(\mathbb{R})$essentially behaving like $u^\beta$, $0<\beta<1$. Precisely for $\chi<1$ andany sufficiently regular initial data $u(x,0)\geq 0$ and $v(x,0)>0$ on$\bar{\Omega}$, we show the existence of global classical solutions. Moreover,if additionally $m:=\int_\Omega u(x,0)$ is sufficiently small, then also theirboundedness is achieved.

Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity