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

TL;DR

This paper proves the global existence and convergence to equilibrium of solutions in a three-dimensional doubly degenerate nutrient taxis system, using novel inequalities to handle the complex diffusion mechanism.

Contribution

It introduces new functional inequalities to establish global solutions and analyze their long-term behavior in a challenging degenerate diffusion system.

Findings

Global weak solutions exist for certain parameter ranges.

Solutions converge to a non-homogeneous equilibrium.

The limiting profile depends on initial data and parameters.

Abstract

The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha}v\nabla v)+\ell uv,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-uv,&x\in \Omega,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^3$ where $\alpha>0,\chi>0$ 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 $\alpha\in(\frac{3}{2},\frac{19}{12})$, 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.