Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source

TL;DR

This paper proves the existence of global weak solutions for a two-dimensional doubly degenerate nutrient taxis system with logistic source, extending previous results and ensuring solutions are continuous and smooth under certain conditions.

Contribution

It establishes the existence of global weak solutions for the system with general degeneracy parameter l, including the case l=2, which aligns with prior specific results.

Findings

Existence of global weak solutions for all regular initial data.

Solutions are continuous in the u-component and smooth in the v-component.

Results include the special case l=2, matching previous studies.

Abstract

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=\Delta v-u v \end{cases} \end{align} in a smooth bounded domain $\Omega \subset \mathbb{R}^2$, where $l \geqslant 1$. It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when $l = 2$, our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)].