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Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source

Zhiguang Zhang, Yuxiang Li

Abstract

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=Δv-u v \end{cases} \end{align} in a smooth bounded domain $Ω\subset \mathbb{R}^2$, where $l \geqslant 1$. It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when $l = 2$, our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)].

Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source

Abstract

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=Δv-u v \end{cases} \end{align} in a smooth bounded domain , where . It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when , our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)].

Paper Structure

This paper contains 5 sections, 18 theorems, 149 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary estimates for the regularized problems
  3. $L^p$ Bounds for $u_{\varepsilon}$
  4. Global existence of weak solutions
  5. Appendix: A key functional inequalities

Key Result

Theorem 1.1

Let $l\geqslant1$ and 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 $u \geqslant 0$ in $\Omega \times(0, \infty)$ and $v>0$ in $\overline{\Omega} \times[0, \infty)$, and that $(u, v)$ solves SYS:MAIN in the sense of Definition def-weak-sol.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 26 more