Global weak solvability for doubly degenerate nutrient taxis system in physical dimension

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 $\alpha\ge0$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system appears to be highly challenging due to the difficulty of quantifying the dissipation of the doubly degenerate diffusive flux, the strong chemotactic effect, particularly as $\alpha$ is close to $2$, and the dimensionally dependent limitation, which is the most difficult one. 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, Y. Li, \textit{Math. Models Methods Appl. Sci.}, 2026] for $\alpha=2$, $N=2$; \item In [G. Li, \textit{J. Differ. Equ.}, 2022] for $\frac{7}{6}<\alpha<\frac{13}{9}$, $N=3$; and \item In [X.M. De-Ji, A. Huang, Y. Wang, \textit{Eur. J. Appl. Math.}] for $\frac{3}{2}<\alpha<\frac{19}{12}$, $N=3$. \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) $0\le\alpha\le1$: Weak chemotaxis effect; (ii) $1<\alpha\le3/2$: Moderate chemotaxis effect; and (iii) $3/2<\alpha<2$: Strong chemotaxis effect.