Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source

In this paper we study the global boundedness of solutions to the fully parabolic attraction-repulsion chemotaxis system with logistic source: ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), vt=Δv−βv+αu, wt=Δw−δw+γu, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥1), where χ, α, ξ, γ, β and δ are positive constants, and f:R→R is a smooth function generalizing the logistic source f(s)=a−bsθ for all s≥0 with a≥0, b>0 and θ≥1. It is shown that when the repulsion cancels the attraction (i.e. χα=ξγ), the solution is globally bounded if n≤3, or θ>θn:=min⁡{n+24,nn2+6n+17−n2−3n+44} with n≥2. Therefore, due to the inhibition of repulsion to the attraction, in any spatial dimension, the exponent θ is allowed to take values less than 2 such that the solution is uniformly bounded in time.