  The taxis-type migration-consumption model accounting for signal-dependent
motilities, as given by \[
  u_t = \Delta \big(u\phi(v)\big),
  v_t = \Delta v-uv,
  \qquad (*) \] is considered for suitably smooth functions $\phi:[0,\infty)\to
R$ which are such that $\phi>0$ on $(0,\infty)$, but that in addition
$\phi(0)=0$ with $\phi'(0)>0$.
  In order to appropriately cope with the diffusion degeneracies thereby
included, this study separately examines the Neumann problem for the linear
equation \[
  V_t = \Delta V + \nabla\cdot \big( a(x,t)V\big) + b(x,t)V \] and establishes
a statement on how pointwise positive lower bounds for nonnegative solutions
depend on the supremum and the mass of the initial data, and on integrability
features of $a$ and $b$.
  This is thereafter used as a key tool in the derivation of a result on global
existence of solutions to (*), smooth and classical for positive times, under
the mere assumption that the suitably regular initial data be nonnegative in
both components. Apart from that, these solutions are seen to stabilize toward
some equilibrium, and as a qualitative effect genuinely due to degeneracy in
diffusion, a criterion on initial smallness of the second component is
identified as sufficient for this limit state to be spatially nonconstant.
