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

This paper is concerned with the doubly degenerate nutrient taxis system $u_t=\mathrm{\nabla }\cdot (u^{l-1}v\mathrm{\nabla }u)-\mathrm{\nabla }\cdot \left(u^{l}v\mathrm{\nabla }v\right)+uv$ and $v_t=\mathrm{\Delta }v-uv$ for some $l⩾1$, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $(n⩽2)$. 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 $l\in [1,\mathrm{\infty })$. Notably, the solution remains uniformly bounded when $l\in [1,\mathrm{\infty })$ in one dimension or $l\in (1,3]$ in two dimensions.