Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption

Abstract

This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: \begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in Ω,\, t>0,\\ & v_{t}=Δv-vw,&x\in Ω,\, t>0,\\ &w_t=Δw-w+u,&x\inΩ,t>0 \end{aligned} \right. \end{equation*} posed on a smooth bounded domain $Ω\subset\mathbb{R}^{3}$ with no-flux boundary conditions. It is shown that for suitably regular initial data $(u_0,v_0,w_0)$, the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0,w_\infty)$ as $t\rightarrow \infty$. Notably, when $u_0$ is nonconstant and the mass of $v_0$ is sufficiently small, the limiting profiles $u_{\infty}$ and $w_{\infty}$ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral $\int_Ωu^{k}v|\nabla u|^2$ with some $k>-1$.