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

Abstract The doubly degenerate nutrient taxis system (0.1) \begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\ \end{aligned} \right . \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^3$ where $\alpha \gt 0,\chi \gt 0$ and $\ell \gt 0$ . 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 $\alpha \in (\frac {3}{2},\frac {19}{12})$ , 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 $(u_\infty , 0)$ as $t\rightarrow \infty$ . Notably, the limiting profile $u_{\infty }$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant.