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This work considers the Keller-Segel consumption system \begin{eqnarray*}\left\{ \begin{array}{llll} u_t=\Delta (u\phi(v))+au-bu^\gamma,\quad &x\in\Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0 \end{array}\right. \end{eqnarray*} in a smoothly bounded domain $\Omega\subset\mathbb{R}^n,$ $n\geq1$, under no-flux boundary conditions, where theparameters $a,b>0$, $\gamma\geq2$, and the motility function $\phi$ suitablygeneralizes the prototype given by $\phi(s)=s^\alpha$ for all $s\geq0$ with$\alpha>0$.  When $\phi$ is appropriately smooth with $\alpha\geq1$, it is shown that ifone of the following cases holds: (i) $\gamma>2$; (ii) $\gamma=2$, either$n\leq2$ or $n\geq3$ and $b$ is sufficiently large, then for all suitablyregular initial data global classical solutions can be constructed. Whereaswhen $\phi$ is considered to be with rather mild regularity properties and$\gamma=2$, for arbitrary $b>0$, this system admits at least one global weaksolution in case $\alpha>0$. In addition, if $\phi$ is suitably smooth with$\alpha>1$, then the above weak solutions become eventually smooth.

Global solutions to a chemotaxis consumption model involving signal-dependent degenerate diffusion and logistic-type dampening