  In this work, we study global existence, eventual smoothness and asymptotical
behavior of positive solutions for a two-species chemotaxis consumption model
in a bounded smooth but not necessarily convex domain $\Omega\subset
\mathbb{R}^n (n=2,3,4,5)$ with nonnegative initial data and homogeneous Neumann
boundary data Under a smallness condition, boundedness of classical solutions
and stabilization to constant equilibrium is known. Here, without any smallness
condition, we show global existence and uniform-in-time boundedness of
classical solutions in 2D and global existence, eventual smoothness and
asymptotical behavior (in convex domains) of weak solutions in nD (n=3,4,5).
Our findings also extend and improve the one-species chemotaxis-consumption
model studied in relevant literature.
