LAUSR

Abstract This paper concerns the existence of bounded classical solutions to the attraction-repulsion chemotaxis system with logistic source (⋆) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ) , x ∈ Ω , t > 0 , 0 = Δ v − β v + α u , x ∈ Ω , t > 0 0 = Δ w − δ w + γ u , x ∈ Ω , t > 0 in a smooth bounded domain Ω ⊆ R N ( N ≥ 1 ) , subject to nonnegative initial data and homogeneous Neumann boundary conditions, where f ( u ) ≤ a − b u r for all u ≥ 0 with some a ≥ 0 , b > 0 and r ≥ 1 . Here χ , α , ξ , β as well as γ and δ are positive constants. It is proved that the corresponding system ( ⋆ ) possesses a unique global bounded classical solution in the balance case χ α = ξ γ with r > 2 N − 2 N or, the attraction domination case ξ α > ξ γ with b ≥ ( N − 2 ) + N ( χ α − ξ γ ) and r = 2 , respectively. The study of this paper improves the results in Li-Xiang (2016) [12] , Xu-Zheng (2018) [39] , Wang (2016) [26] as well as Zhao et al. (2017) [42] and Tello-Winkler (2007) [23] , in which, the assumption b > ( N − 2 ) + N ( χ α − ξ γ ) (see Li-Xiang (2016) as well as Wang (2016) and Zhao et al. (2017)) or b > ( N − 2 ) + N χ (see Tello-Winkler (2007)) or r > 2 N + 2 N + 2 (see Xu-Zheng (2018)) or r > N 2 + 4 N − N 2 (see Li-Xiang (2016)) are intrinsically required.