Global boundedness in an attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanism

We study the following chemotaxis system involving nonlinear indirect signal mechanism{ut=Δu−ξ∇⋅(u∇v)+χ∇⋅(u∇z),x∈Ω,t>0,vt=Δv−v+wγ1,x∈Ω,t>0,0=Δw−w+uγ2,x∈Ω,t>0,0=Δz−z+uγ3,x∈Ω, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn(n≥1), where ξ,χ,γ1,γ2,γ3>0. With the aid of maximum Sobolev regularity and a priori estimates of w and z, it has been proved that if either random diffusion or repulsion mechanism dominates the nonlinear indirect attraction mechanism with γ1γ2