This paper considers the following coupled chemotaxis system:
⎧⎪⎨⎪⎩
ut = ∇ · (φ(u)∇u) – χ1∇ · (u∇w) + μ1u(1 – u – a1v),
vt = ∇ · (ψ(v)∇v) – χ2∇ · (v∇w) + μ2v(1 – a2u – v),
wt = w – γ w + αu + βv,
with homogeneous Neumann boundary conditions in a bounded domain ⊂ RN
(N ≥ 3) with smooth boundaries, where χ1, χ2, μ1, μ2, a1, a2, α, β and γ are positive.
Based on the maximal Sobolev regularity, the existence of a unique global bounded
classical solution of the problem is established under the assumption that both μ1
and μ2 are sufficiently large.

• Boundedness in a quasilinear fully parabolic two-species chemotaxis system of higher dimension