Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption

This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: \begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in Ω,\, t>0,\\ & v_{t}=Δv-vw,&x\in Ω,\, t>0,\\ &w_t=Δw-w+u,&x\inΩ,t>