Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system

This paper deals with an attraction-repulsion chemotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,τ1vt=Δv+αu−βv,x∈Ω,t>0,τ2wt=Δw+γu−δw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN (N≥2), where parameters τi(i=1,2), χ, ξ, α, β, γ and δ are positive, and diffusion coefficient D(u)∈C2(0,+∞) satisfies D(u)>0 for u≥0, D(u)≥dum−1 with d>0 and m≥1 for all u>0. It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution for m>2−2N. In particular in the case τ1=τ2 and χα=ξγ, the solution is globally bounded if m>2−2N−N+2N2−N+2. Therefore, due to the inhibition of repulsion to the attraction, the range of m>2−2N of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction-repulsion chemotaxis system with nonlinear diffusion.