A parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source

This paper deals with the parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source{ut=∇⋅(D(u)∇u)−∇⋅(χu∇v)+∇⋅(ξu∇w)+ru−μu2,x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0, under no-flux boundary conditions in bounded domain with smooth boundary, where χ,ξ,α,β,γ,δ,r and μ are assumed to be positive. When Ω⊆R3, D(u) is assumed to satisfy D(0)>0,D(u)≥cDum−1withm≥1andcD>0, it is proved that if χα−ξγ>0 and μ=13(χα−ξγ), then for any given u0∈W1,∞(Ω), the system possesses a global and bounded classical solution. For the case where D(u)≡1 and n≥3, the convergence rate of the solution is established. When the random motion of the chemotactic species is neglected i.e. (D(u)≡0) and Ω⊂Rn(n≥2) is a convex domain, boundedness and the finite time blow up of the solution are investigated.