Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system

The doubly degenerate nutrient taxis system (0.1) {equation} \{ {aligned} &uₜ=∇ · (uv∇ u)-χ ∇ · (uαv∇ v)+ uv,&x∈ Ω ,\, t> 0,\\[5pt] & vₜ=Δ v-uv,&x∈ Ω ,\, t> 0,\\ {aligned} . {equation} is considered under zero-flux boundary conditions in a smoothly bounded domain Ω ⊂ R³ where α > 0,χ > 0 and > 0 . By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in (0.1), it is shown that for α ∈ (3/2,19/12) , the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium (u_∞ , 0) as t→ ∞ . Notably, the limiting profile u∞ is non-homogeneous when the initial signal concentration v₀ is sufficiently small, provided the initial data u₀ is not identically constant.