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

Abstract

In a smoothly bounded convex domain $\Omega \subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=Δ\big(uϕ(v)\big), v_t=Δv-uv, \end{array} \right. \] is considered under the assumption that near the origin, the function $\varphi$ suitably generalizes the prototype given by \[ ϕ(ξ)=ξ^α, \qquad ξ\in [0,ξ_0]. \] 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 $C(T):=\mathrm{esssup}{}_{t\in (0,T)}{\int }_{\Omega }u(\cdot ,t)\ln u(\cdot ,t)<\infty$ for all $T>0$, with ${\sup }_{T>0}C(T)<\infty$ if $\alpha \in [1,2]$.