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Global boundedness for a two-dimensional doubly degenerate nutrient taxis system

Zhiguang Zhang, Yuxiang Li

Abstract

This paper is concerned with the doubly degenerate nutrient taxis system $u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ uv$ and $v_t=Δv-u v$ for some $l \geqslant 1$, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset \mathbb{R}^n$ $(n \leqslant 2)$. Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if $l \in[1,3]$, then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for $l \in[1,\infty)$. Notably, the solution remains uniformly bounded when $l \in[1,\infty)$ in one dimension or $l \in(1,3]$ in two dimensions.

Global boundedness for a two-dimensional doubly degenerate nutrient taxis system

Abstract

This paper is concerned with the doubly degenerate nutrient taxis system and for some , subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain . Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if , then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for . Notably, the solution remains uniformly bounded when in one dimension or in two dimensions.

Paper Structure

This paper contains 8 sections, 40 theorems, 263 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Fundamental a Priori Estimates for \ref{['sys-regul']}
  4. The Two-Dimensional Case
  5. Uniform estimates in Lebesgue spaces
  6. Higher-Order Estimates. Proof of Theorem \ref{['thm-1.1']}
  7. Proof of Theorem \ref{['thm-1.1xzc']}
  8. The One-Dimensional Case. Proof of Theorem \ref{['thm-1.1a']}

Key Result

Theorem 1.1

Let $1<l\leqslant3$ and let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain with smooth boundary. Assume that the initial value $\left(u_0, v_0\right)$ satisfies assIniVal. Then there exist functions such that $u > 0$ a.e 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, and that $(

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 59 more