New results on the global dynamics for a two-species chemotaxis system with two chemicals and logistic sources

This paper deals with the following competitive two-species chemotaxis system with two chemicals with a source term of logistic type ut=Δu−χ1∇⋅(u∇υ)+η1u−μ1u2, υt=Δυ−υ+ω, ωt=Δω−χ2∇⋅(ω∇z)+η2ω−μ2ω2, 0=Δz−z+u, (x,t)∈Ω×(0,T) under homogeneous Neumann boundary conditions [Formula presented] and initial conditions u(x,0)=u0(x), υ(x,0)=υ0(x), ω(x,0)=ω0(x), x∈Ω, in a bounded domain Ω⊂RN (N≥1), where the parameters χ1, μ1, μ2>0 and η1, η2, χ2∈R. It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds: (i)χ2≤0, (ii)χ2>0 as well as μ2 is large enough, which partially improves the results of [25].