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

The doubly degenerate nutrient taxis system (Formula presented) is considered under zero-flux boundary conditions in a smoothly bounded domain Ω⊂R³ where α > 0, χ > 0 and L > 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 (Formula presented), 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.