A strongly degenerate migration-consumption model in domains of arbitrary dimension

In a smoothly bounded convex domain $\mathrm{\Omega }\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }(u\phi (v)),v_t=\mathrm{\Delta }v-uv,\end{array}$$ is considered under the assumption that near the origin, the function $\phi$ suitably generalizes the prototype given by $$\phi (\xi )={\xi }^{\alpha },\xi \in [0,{\xi }_0]\mathrm{.}$$ By means of separate approaches, it is shown that in both cases $\alpha \in (0,1)$ and $\alpha \in [1,2]$ some global weak solutions exist which, inter alia, satisfy $C(T):=\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{p}{}_{t\in (0,T)}{\int }_{\mathrm{\Omega }}u(\cdot ,t)\ln u(\cdot ,t)<\mathrm{\infty }$ for all $T>0$, with ${\sup }_{T>0}C(T)<\mathrm{\infty }$ if $\alpha \in [1,2]$.