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

Abstract

The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-χ\nabla \cdot (u^αv\nabla v)+\ell uv,&x\in Ω,\, t>0,\\ & v_{t}=Δv-uv,&x\in Ω,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ where $\hat{I}\pm >0,\ddot{I}‡>0$ and $\ell >0$. By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $\hat{I}\pm \in (\frac{3}{2},\frac{19}{12})$, 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 $t\to \infty$. 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.

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