Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source

This paper deals with an attraction–repulsion chemotaxis system with logistic source u_t=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,v_t=Δv+αu−βv,x∈Ω,t>0,w_t=Δw+γu−δw,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R^N (N≥1), where parameters χ, ξ, α, β, γ and δ are positive and f(s)=κs−μs^1+kwithκ∈R,μ>0andk≥1. It is shown that the corresponding system possesses a unique global bounded classical solution in the cases k>1 or k=1 with μ>C_Nμ^∗ for some μ^∗,C_N>0. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when κ<0 (resp. κ=0), the corresponding solution of the system decays to (0,0,0) exponentially (resp. algebraically), and when κ>0 the solution converges to κμ^1∕k,αβκμ^1∕k,γδκμ^1∕k exponentially if μ is larger.