A degenerate migration-consumption model in domains of arbitrary dimension

Abstract

In a smoothly bounded convex domain Ω⊂Rn ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem forut=Δuϕ(v),vt=Δv−uv, $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given byϕ(ξ)=ξα,ξ∈[0,ξ0]. $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ 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, satisfyC(T)≔ess supt∈(0,T)∫Ωu(⋅,t)ln⁡u(⋅,t)0, $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){}0,$$ with supT>0 C(T) < ∞ if α ∈ [1, 2].

Keywords

• chemotaxis
• degenerate diffusion
• a priori estimate
• 35k65 (primary)
• 35k51
• 35q92
• 92c17
• 35k57 (secondary)