LAUSR

Abstract This paper studies the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ) , 0 = Δ v + α u k − β v , 0 = Δ w + γ u l − δ w , in bounded domain Ω ⊂ R n , n ≥ 1 , subject to the non-flux boundary conditions, where the nonlinear productions for the attraction and repulsion chemicals are described via α u k and γ u l respectively, and the logistic source f ∈ C 2 [ 0 , ∞ ) satisfying f ( u ) ≤ u ( a − b u s ) with s > 0 , f ( 0 ) ≥ 0 . It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with max ⁡ { l , s , 2 n } > k , the solutions would be globally bounded. Furthermore, under the three balance situations, namely, k = s > l , k = l > s or k = s = l , the boundedness of solutions depends on the sizes of the coefficients involved. This extends the global boundedness criteria established by Zhang and Li (2016) [20] for the attraction-repulsion chemotaxis system with linear productions and logistic source.