Asymptotic stability of the one-dimensional attraction-repulsion chemotaxis system

The asymptotic stability of the one-dimensional attraction-repulsion chemotaxis system is concerned with in this paper. First of all, we get the globally bounded classical solutions of the system with initial data ( u 0 , v 0 , ω 0 ) ∈ ( W 1 , ∞ ( Ω ) ) 3 . Next, by constructing an appropriate Lyapunov function, we also show that the solutions converge exponentially to the constant steady state ( u ̵̄ 0 , α β u ̵̄ 0 , γ δ u ̵̄ 0 ) if ξγ χα > max { β δ , δ β } . It is worth mentioning that Jin and Wang (2015) showed that the solutions converge algebraically to the constant steady state under conditions β = δ and ξγ − χα 0.