  We consider the Keller-Segel-type migration-consumption system involving
signal-dependent motilities, $$\left\{ \begin{array}{l}
  u_t = \Delta \big(u\phi(v)\big), \\[1mm]
  v_t = \Delta v-uv,
  \end{array} \right.
  \qquad \qquad$$ in smoothly bounded domains $\Omega\subset\mathbb{R}^n$,
$n\ge 1$. Under the assumption that $\phi\in C^1([0,\infty))$ is positive on
$[0,\infty)$, and for nonnegative initial data from
$(C^0(\overline{\Omega}))^\star \times L^\infty(\Omega)$, previous literature
has provided results on global existence of certain very weak solutions with
possibly quite poor regularity properties, and on large time stabilization
toward semitrivial equilibria with respect to the topology in
$(W^{1,2}(\Omega))^\star \times L^\infty(\Omega)$.
  The present study reveals that solutions in fact enjoy significantly stronger
regularity features when $0<\phi\in C^3([0,\infty))$ and the initial data
belong to $(W^{1,\infty}(\Omega))^2$: It is firstly shown, namely, that then in
the case $n\le 2$ an associated no-flux initial-boundary value problem even
admits a global classical solution, and that each of these solutions smoothly
stabilizes in the sense that as $t\to\infty$ we have $$ \begin{align*}
  u(\cdot,t) \to \frac{1}{|\Omega|}\int_\Omega u_0
  \qquad \text{ and } \qquad
  v(\cdot,t)\to 0
  \qquad \qquad (\star) \end{align*}$$ even with respect to the norm in
$L^\infty(\Omega)$ in both components.
  In the case when $n\ge 3$, secondly, some genuine weak solutions are found to
exist globally, inter alia satisfying $\nabla u\in
L^\frac{4}{3}_{loc}(\overline{\Omega}\times [0,\infty);\mathbb{R}^n)$. In the
particular three-dimensional setting, any such solution is seen to become
eventually smooth and to satisfy ($\star$).
