  In this paper, we deal with quasilinear Keller--Segel systems with indirect
signal production, $$\begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u) -
\nabla \cdot (u \nabla v), &x \in \Omega,\ t> 0,\\ 0 = \Delta v - \mu(t) + w,
&x \in \Omega,\ t> 0,\\ w_t + w = u, &x \in \Omega,\ t> 0, \end{cases}$$
complemented with homogeneous Neumann boundary conditions and suitable initial
conditions, where $\Omega\subset\mathbb R^n$ $(n\ge3)$ is a bounded smooth
domain, $m\ge1$ and $$\mu(t) := \frac{1}{|\Omega|} w(\cdot, t)
\qquad\mbox{for}\ t>0.$$ We show that in the case $m\ge2-\frac{2}{n}$, there
exists $M_c>0$ such that if either $m>2-\frac{2}{n}$ or $\int_\Omega u_0 <M_c$,
then the solution exists globally and remains bounded, and that in the case
$m\le2-\frac{2}{n}$, if either $m<2-\frac{2}{n}$ or
$M>2^\frac{n}{2}n^{n-1}\omega_n$, then there exist radially symmetric initial
data such that $\int_\Omega u_0 = M$ and the solution blows up in finite or
infinite time, where the blow-up time is infinite if $m=2-\frac2n$. In
particular, if $m=2-\frac{2}{n}$ there is a critical mass phenomenon in the
sense that $\inf\left\{M > 0 : \exists u_0 \text{ with } \int_\Omega u_0 = M
\text{ such that the corresponding solution blows up in infinite time}\right\}$
is a finite positive number.
