具有间接消耗的三维双退化营养系统的空间非均匀性

中文摘要 AI 辅助

本文研究一类具有间接消耗的双退化营养趋化系统的全局动力学，该系统定义在三维欧氏空间中光滑有界区域Ω上，配备无通量边界条件，其方程组为：\begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in \Omega,\\, t>0,\\\\ & v_{t}=\Delta v-vw,&x\in \Omega,\\, t>0,\\\\ &w_t=\Delta w-w+u,&x\in\Omega,t>0 \end{aligned} \right. \end{equation*} 研究表明，对于适当正则的初始数据$(u_0,v_0,w_0)$，对应的初边值问题存在全局弱解；进一步，在合适的拓扑框架下，该解当$t\rightarrow \infty$时收敛到平衡态$(u_\infty, 0,w_\infty)$。值得注意的是，当$u_0$非恒定且$v_0$的质量足够小时，极限剖面$u_\infty$和$w_\infty$呈空间非均匀性，捕捉到营养匮乏环境中涌现的斑图。本分析的核心是引入新型函数不等式，为积分$\int_{\Omega}u^{k}v|\nabla u|^2$（其中$k>-1$）提供下界估计。

英文摘要

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 $k>-1$.

发表机构

• School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
• College of Science, Minzu University of China（中央民族大学理学院）

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