Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system

This paper deals with an attraction-repulsion chemotaxis system with logistic source {ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),(x,t)∈Ω×(0,∞),vt=Δv−α1v+β1u,(x,t)∈Ω×(0,∞),wt=Δw−α2w+β2u,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥3) with nonnegative initial data (u0,v0,w0)∈W1,∞(Ω)×W2,∞(Ω)×W2,∞(Ω), where χ>0, ξ>0, αi>0, βi>0(i=1,2) and f(u)≤au−μu2 with a≥0 and μ>0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n≥3, α1=α2 and there exists θ0>0 such that χβ1+ξβ2μ<θ0. The main aim of this paper is to solve the higher-dimensional boundedness question addressed by Xie and Xiang in [IMA J. Appl. Math. 81 (2016) 165-198].