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Boundedness in a two-dimensional doubly degenerate nutrient taxis system with logistic source

Zhiguang Zhang, Yuxiang Li

TL;DR

This work addresses the global behavior of a two-dimensional doubly degenerate nutrient-taxis system with logistic source, proving the existence of a global bounded weak solution for arbitrary smooth initial data in a bounded smooth domain under Neumann boundary conditions. The authors introduce a regularized problem with a small parameter $\varepsilon$, and develop a weighted energy framework that couples an entropy-like term for $u$ with a higher-order energy for $v$, yielding uniform-in-$\varepsilon$ controls. They establish uniform $L^p$ bounds for all $p>1$ and derive boundedness in $L^{\infty}$ for $u_\varepsilon$ and $W^{1,\infty}$ control for $v_\varepsilon$, which together enable compactness and passage to the limit to obtain a global bounded weak solution $(u,v)$ of the original system. This result extends previous 1D and restricted 2D findings to general planar domains and highlights the dissipative effect of the logistic term in ensuring global regularity for a biologically inspired aggregation model.

Abstract

We are concerned with the following doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq-0.1} u_t=\nabla\cdot(u v\nabla u)-\nabla\cdot(u^{2} v\nabla v)+u-u^2,\\[1mm] v_t=Δv-u v, \end{cases} \end{align} posed in a bounded smooth domain $Ω\subset\mathbb{R}^2$ under homogeneous Neumann boundary conditions. This model was introduced to describe the aggregation patterns of colonies of \emph{Bacillus subtilis} observed on thin agar plates. Previous results have established global boundedness in one space dimension and, in two dimensions, under additional assumptions such as small initial data or convex domains (see, e.g., M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021; M. Winkler, \textit{J. Differ. Equ.}, 2024). In the presence of the quadratic degradation term in the logistic growth, which markedly enhances the dissipative structure of the system, and by employing a weighted energy method, we prove that for arbitrary smooth initial data the problem \eqref{eq-0.1} admits a global weak solution that remains uniformly bounded in time.

Boundedness in a two-dimensional doubly degenerate nutrient taxis system with logistic source

TL;DR

This work addresses the global behavior of a two-dimensional doubly degenerate nutrient-taxis system with logistic source, proving the existence of a global bounded weak solution for arbitrary smooth initial data in a bounded smooth domain under Neumann boundary conditions. The authors introduce a regularized problem with a small parameter , and develop a weighted energy framework that couples an entropy-like term for with a higher-order energy for , yielding uniform-in- controls. They establish uniform bounds for all and derive boundedness in for and control for , which together enable compactness and passage to the limit to obtain a global bounded weak solution of the original system. This result extends previous 1D and restricted 2D findings to general planar domains and highlights the dissipative effect of the logistic term in ensuring global regularity for a biologically inspired aggregation model.

Abstract

We are concerned with the following doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{}\label{eq-0.1} u_t=\nabla\cdot(u v\nabla u)-\nabla\cdot(u^{2} v\nabla v)+u-u^2,\\[1mm] v_t=Δv-u v, \end{cases} \end{align} posed in a bounded smooth domain under homogeneous Neumann boundary conditions. This model was introduced to describe the aggregation patterns of colonies of \emph{Bacillus subtilis} observed on thin agar plates. Previous results have established global boundedness in one space dimension and, in two dimensions, under additional assumptions such as small initial data or convex domains (see, e.g., M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021; M. Winkler, \textit{J. Differ. Equ.}, 2024). In the presence of the quadratic degradation term in the logistic growth, which markedly enhances the dissipative structure of the system, and by employing a weighted energy method, we prove that for arbitrary smooth initial data the problem \eqref{eq-0.1} admits a global weak solution that remains uniformly bounded in time.
Paper Structure (5 sections, 23 theorems, 145 equations)

This paper contains 5 sections, 23 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local existence and basic properties of solutions.
  4. Uniform $L^p$ Estimates for $u_{\varepsilon}$ with $p>1$
  5. Uniform $L^\infty \times W^{1,\infty}$ Boundedness of $(u_\varepsilon, v_\varepsilon)$

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with smooth boundary. Assume that the initial data $\left(u_0, v_0\right)$ satisfies assIniVal. Then there exist functions such that $u \geqslant 0$ and $v>0$ in $\bar{\Omega} \times[0, \infty)$, and that $(u, v)$ solves SYS:MAIN in the sense of Definition def-weak-sol, and that $(u, v)$ is bounded in the sense that where $C>0$ is a constant

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.1
  • Lemma 2.1: 2014-SJMA-StinnerSurulescuWinkler
  • Lemma 2.2: 1997--Temam
  • Lemma 2.3: 2018-MMMAS-PangWang
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 32 more