Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities

Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad (\star)$$ are studied in smoothly bounded domains $\Omega\subset\mathbb{R}^n$, $n\ge 1$: It is shown that if $\phi\in C^0([0,\infty))$ is positive on $[0,\infty)$, then for any initial data $(u_0,v_0)$ belonging to the space $(C^0(\overline{\Omega}))^\star\times L^\infty(\Omega)$ an associated no-flux type initial-boundary value problem admits a global very weak solution. Beyond this initial relaxation property, it is seen that under the additional hypotheses that $\phi\in C^1([0,\infty))$ and $n\le 3$, each of these solutions stabilizes toward a semi-trivial spatially homogeneous steady state in the large time limit. By thus applying to irregular and partially even measure-type initial data of arbitrary size, this firstly extends previous results on global solvability in ($\star$) which have been restricted to initial data not only considerably more regular but also suitably small. Secondly, this reveals a significant difference between the large time behavior in ($\star$) and that in related degenerate counterparts involving functions $\phi$ with $\phi(0)=0$, about which, namely, it is known that some solutions may asymptotically approach nonhomogeneous states.