Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system
De-Ji-Xiang-Mao, Ai Huang, Yifu Wang
TL;DR
This work analyzes a three-dimensional zero-flux, doubly degenerate nutrient taxis system and proves global existence of a continuous weak solution for $\alpha$ in $(\tfrac{3}{2},\tfrac{19}{12})$ on smoothly bounded domains, without requiring convexity. The authors develop a novel energy framework and functional inequalities to handle diffusion degeneracy and taxis, enabling a bootstrap of $L^{p}$ bounds for $u$ and leading to strong a priori estimates that allow passage to the limit in a regularized problem. They establish large-time behavior: solutions converge to an equilibrium $(u_\infty,0)$ with $u_\infty\in L^{p}$ for all $p\ge1$, and show that $u_\infty$ can be spatially nonuniform when the initial nutrient signal $v_0$ is small and $u_0$ is not constant. A key theme is handling non-convex 3D domains and obtaining nontrivial asymptotics through duality estimates, detailed energy analysis, and refined interpolation inequalities, highlighting the destabilizing potential of taxis even under dissipation.
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 $Ω\subset\mathbb{R}^3$ where $α>0,χ>0$ 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 $α\in(\frac{3}{2},\frac{19}{12})$, 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.