Global boundedness in an attraction–repulsion chemotaxis system with logistic source

Abstract

We study the attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source: u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ) , 0 = Δ v − β v + α u , 0 = Δ w − δ w + γ u , subject to the non-flux boundary conditions in a bounded domain Ω ⊂ R n ( n ≥ 1 ) with smooth boundary, f ( s ) ≤ a − b s η for all s ≥ 0 , where constants χ , ξ , η , α , δ , γ , b > 0 , a ≥ 0 . The global boundedness of solutions to this problem was established by Li and Xiang (2016) for the repulsion domination case χ α < ξ γ with η ≥ 1 , the attraction domination case χ α > ξ γ with η > 2 (or η = 2 , b properly large), and the balance case χ α = ξ γ with η > 1 2 ( n 2 + 4 n − n + 2 ) , respectively. In the present paper we prove for the balance case χ α = ξ γ that the weakened restriction η > 2 n + 2 n + 2 is sufficient to ensure the global boundedness of solutions.