Global boundedness for a two-dimensional doubly degenerate nutrient taxis system

TL;DR

This paper proves the global boundedness and existence of solutions for a two-dimensional doubly degenerate nutrient taxis system under certain conditions, extending previous results to broader parameter ranges.

Contribution

It establishes global weak solutions and uniform boundedness for the system in 2D for $l \\in[1,3]$ and in 1D for all $l \\geq 1$, using novel analytical approaches.

Findings

Global weak solutions exist for $l \\in[1,3]$ in 2D.

Solutions are uniformly bounded for $l \\in[1,\\infty)$ in 1D.

Solutions are uniformly bounded for $l \\in(1,3]$ in 2D.

Abstract

This paper is concerned with the doubly degenerate nutrient taxis system $u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ uv$ and $v_t=\Delta v-u v$ for some $l \geqslant 1$, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega \subset \mathbb{R}^n$ $(n \leqslant 2)$. Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if $l \in[1,3]$, then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for $l \in[1,\infty)$. Notably, the solution remains uniformly bounded when $l \in[1,\infty)$ in one dimension or $l \in(1,3]$ in two dimensions.