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This paper deals with a two-species chemotaxis system  ut = ∇ · (D1(u)∇u)−∇ · (uχ1(w)∇w) + μ1u(1− u− a1v), x ∈ Ω, t > 0, vt = ∇ · (D2(v)∇v)−∇ · (vχ2(w)∇w) + μ2v(1− a2u− v), x ∈ Ω, t > 0, wt = ∆w− (αu + βv)w, x ∈ Ω, t > 0, where Ω ⊂ Rn (n ≥ 1) is a bounded domain with smooth boundary ∂Ω; χi(i = 1, 2) are chemotactic functions satisfying χ′ i ≥ 0; the parameters μ1, μ2 > 0, a1, a2 > 0 and α, β > 0, the initial data (u0, v0) ∈ (C0(Ω))2 and w0 ∈ W1,∞(Ω) are non-negative. Based on the maximal Sobolev regularity, it is shown that this system possesses a unique global bounded classical solution provided that the logistic growth coefficients μ1 and μ2 are sufficiently large.

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant