Large time behavior of solutions to a fully parabolic attraction–repulsion chemotaxis system with logistic source. (February 2018)

• Record Type: Journal Article
• Authors: Li, Jing; Ke, Yuanyuan; Wang, Yifu
• Is Part Of: Nonlinear analysis. Volume 39(2018)
• Page Start: 261
• Page End: 277
• Publication Date: 2018-02
• Subjects: Chemotaxis -- Attraction–repulsion -- Boundedness asymptotic behavior; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.07.002
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: This paper deals with an attraction–repulsion chemotaxis system with logistic source u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ), x ∈ Ω, t > 0, v t = Δ v + α u − β v, x ∈ Ω, t > 0, w t = Δ w + γ u − δ w, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N ( N ≥ 1 ), where parameters χ, ξ, α, β, γ and δ are positive and f ( s ) = κ s − μ s 1 + k w i t h κ ∈ R, μ > 0 a n d k ≥ 1 . It is shown that the corresponding system possesses a unique global bounded classical solution in the cases k > 1 or k = 1 with μ > C N μ ∗ for some μ ∗, C N > 0 . Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when κ < 0 (resp. κ = 0 ), the corresponding solution of the system decays to ( 0, 0, 0 ) exponentially (resp. algebraically), and when κ > 0 the solution converges to κ μ 1 ∕ k, α β κ μ 1 ∕ k, γ δ κ μ 1 ∕ k exponentially if μ is larger.