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

This paper deals with an attraction–repulsion chemotaxis system {u_t=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,τ₁v_t=Δv+αu−βv,x∈Ω,t>0,τ₂w_t=Δw+γu−δw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R^N (N≥2), where parameters τ_i(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)≥du^m−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−[Formula presented]. In particular in the case τ₁=τ₂ and χα=ξγ the solution is globally bounded if m>2−[Formula presented]−[Formula presented]. Therefore, due to the inhibition of repulsion to the attraction, the range of m>2−[Formula presented] of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction–repulsion chemotaxis system with nonlinear diffusion.