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

Abstract The doubly degenerate nutrient taxis system (0. 1) equation \ aligned &uₓ= (uv u) - (u^{ v v) + uv, &x, \, t 0, \\5pt & vₓ= v-uv, &x, \, t 0, \\ aligned. equation is considered under zero-flux boundary conditions in a smoothly bounded domain R³ where 0, 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 (32, 1912), 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.