Title: Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension Show Chinese title

Title: 双退化营养趋化系统在物理维数中的广泛细菌响应全局可解性

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. Show Chinese abstract

Abstract: 受细菌对环境条件响应研究的启发，我们考虑了双退化营养趋化系统\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*}，在无通量边界条件和光滑初始数据下，其中$\alpha\in\mathbb{R}$是细菌响应参数。由于双重非线性扩散及其退化性以及强烈的化学趋向效应，该趋化系统的弱解的全局可解性极具挑战性，其中后者在物种密度较大时如果$\alpha$接近$2$会非常强烈。关于所考虑系统全局弱可解性的最新研究成果总结如下\begin{itemize} \item 在 [M. Winkler,\textit{交易 美国 数学 社会}, 2021] 中对于$\alpha=2$,$N=1$; \item 在 [M. Winkler,\textit{J. 差分方程}, 2024] 中，对于$1\le\alpha\le 2$,$N=2$，当初始数据的大小较小时如果$\alpha=2$; \item 在 [Z. Zhang 和 Y. Li,\textit{arXiv:2405.20637}, 2024] 中对于$\alpha=2$,$N=2$; 以及 \item 在 [G. Li,\textit{J. 差分方程}, 2022] 中对于$\frac{7}{6}<\alpha<\frac{13}{9}$,$N=3$。 \end{itemize}。我们的工作旨在提供在物理维度设置下$0\le\alpha<2$的全局弱可解性的图景$N=3$。 如分析所建议的，它分为三个可分离的情况，包括（i）$0\le\alpha\le 1$：弱趋化效应；（ii）$1<\alpha\le 3/2$：中等趋化效应；和（iii）$3/2<\alpha<2$：强趋化效应。