Boundedness in a 4-dimensional attraction-repulsion chemotaxis system

In this paper we deal with the following attraction-repulsion chemotaxis system { u t = ∆ u − χ ∇ · ( u ∇ v )+ ξ ∇ · ( u ∇ w ) , x ∈ Ω , t > 0 , 0 = ∆ v − βv + αu , x ∈ Ω , t > 0 , 0 = ∆ w − δw + γu , x ∈ Ω , t > 0 , ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 )= u 0 ( x ) , x ∈ Ω , under homogenous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R 4 , where χ , ξ , β , α , δ and γ are positive constants. In this paper, we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption ξγ = χα and ξ δ λ 0 γ ∫ Ω u 0 < 1 C GN , where C GN and λ 0 are some positive constants only depending on Ω. This result significantly improves or extends previous results of several authors (see Remark 1.1).