A degenerate migration-consumption model in domains of arbitrary dimension

Abstract

In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t = Δ v − u v , $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ ( ξ ) = ξ α , ξ ∈ [ 0 , ξ 0 ] . $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy C ( T ) ≔ ess sup t ∈ ( 0 , T ) ∫ Ω u ( ⋅ , t ) ln ⁡ u ( ⋅ , t ) < ∞ for all T > 0 , $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$ with sup T>0 C(T) < ∞ if α ∈ [1, 2].