Boundedness in a two-dimensional doubly degenerate nutrient taxis system with logistic source

TL;DR

This paper proves that a two-dimensional nutrient taxis system with logistic growth and quadratic degradation admits globally bounded weak solutions for arbitrary smooth initial data, extending previous results under more general conditions.

Contribution

It demonstrates global boundedness of solutions in 2D for a complex taxis system with logistic source, using a weighted energy method, without restrictions on initial data or domain shape.

Findings

Global weak solutions exist for all smooth initial data.

Solutions remain uniformly bounded over time.

The quadratic degradation term enhances system dissipativity.

Abstract

We are concerned with the following doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq-0.1} u_t=\nabla\cdot(u v\nabla u)-\nabla\cdot(u^{2} v\nabla v)+u-u^2,\\[1mm] v_t=\Delta v-u v, \end{cases} \end{align} posed in a bounded smooth domain $\Omega\subset\mathbb{R}^2$ under homogeneous Neumann boundary conditions. This model was introduced to describe the aggregation patterns of colonies of \emph{Bacillus subtilis} observed on thin agar plates. Previous results have established global boundedness in one space dimension and, in two dimensions, under additional assumptions such as small initial data or convex domains (see, e.g., M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021; M. Winkler, \textit{J. Differ. Equ.}, 2024). In the presence of the quadratic degradation term in the logistic growth, which markedly enhances the dissipative structure of the system, and by employing a weighted energy method, we prove that for arbitrary smooth initial data the problem \eqref{eq-0.1} admits a global weak solution that remains uniformly bounded in time.