  In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a
no-flux initial-boundary value problem for \[
  \left\{ \begin{array}{l}
  u_t=\Delta \big(u\phi(v)\big),
  v_t=\Delta v-uv,
  \end{array} \right. \] is considered under the assumption that near the
origin, the function $\phi$ suitably generalizes the prototype given by \[
  \phi(\xi)=\xi^\alpha,
  \qquad \xi\in [0,\xi_0]. \] 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 $C(T):= {\rm esssup} {}_{t\in (0,T)}
\int_\Omega u(\cdot,t)\ln u(\cdot,t) < \infty$ for all $T>0$, with $\sup_{T>0}
C(T)<\infty$ if $\alpha\in [1,2]$.
