Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension

TL;DR

This paper investigates the global existence of weak solutions for a complex bacterial response model in three dimensions, covering a wide range of chemotactic sensitivities and extending previous results to new parameter regimes.

Contribution

It establishes the global solvability of weak solutions for the nutrient taxis system in three dimensions for all response parameters between 0 and 2, filling gaps in existing literature.

Findings

Global weak solutions exist for 0 ≤ α ≤ 1 with weak chemotaxis.

Solutions are also shown for 1 < α ≤ 3/2 with moderate chemotaxis.

The study covers the challenging case 3/2 < α < 2 with strong chemotaxis.

Abstract

Motivated by the study of bacteria's response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begin{align*} \begin{cases} u_t=\nabla\cdot(uv\nabla u)-\chi\nabla\cdot(u^{\alpha}v\nabla v)+\ell uv,\\ v_t=\Delta v-uv, \end{cases} \end{align*} subjected to no-flux boundary conditions and smooth initial data, where $\alpha\in\mathbb{R}$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system is highly challenging due to not only the doubly nonlinear diffusion and its degeneracy but also the strong chemotactic effect, where the latter is strong at the large species density if $\alpha$ is close to $2$. Recent findings on the global weak solvability for the considered system are summarised as follows \begin{itemize} \item In [M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021] for $\alpha=2$, $N=1$; \item In [M. Winkler, \textit{J. Differ. Equ.}, 2024] for $1\le\alpha\le 2$, $N=2$ with initial data of small size if $\alpha=2$; \item In [Z. Zhang and Y. Li, \textit{arXiv:2405.20637}, 2024] for $\alpha=2$, $N=2$; and \item In [G. Li, \textit{J. Differ. Equ.}, 2022] for $\frac{7}{6}<\alpha<\frac{13}{9}$, $N=3$. \end{itemize} Our work aims to provide a picture of global weak solvability for $0\le\alpha<2$ in the physically dimensional setting $N=3$. As suggested by the analysis, it is divided into three separable cases, including (i) $0\le\alpha\le 1$: Weak chemotaxis effect; (ii) $1<\alpha\le 3/2$: Moderate chemotaxis effect; and (iii) $3/2<\alpha<2$: Strong chemotaxis effect.