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

Abstract

In a smoothly bounded convex domain $Ω\subset R^n$ 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 $ϕ$ 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 $α\in (0,1)$ and $α\in [1,2]$ some global weak solutions exist which, inter alia, satisfy $C(T):= {\rm esssup} {}_{t\in (0,T)} \int_Ωu(\cdot,t)\ln u(\cdot,t) < \infty$ for all $T>0$, with $\sup_{T>0} C(T)<\infty$ if $α\in [1,2]$.