Large time behavior of solution to an attraction–repulsion chemotaxis system with logistic source in three dimensions

• Published in: Journal of mathematical analysis and applications Vol. 448; no. 2; pp. 914 - 936
• Main Authors: Li, Dan, Mu, Chunlai, Lin, Ke, Wang, Liangchen
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2017
• Subjects: Asymptotic behavior; Attraction–repulsion; Boundedness; Chemotaxis; Logistic source; Logistic source; Chemotaxis; Boundedness; Asymptotic behavior; Attraction–repulsion
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2016.11.036

This paper studies the attraction–repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), vt=Δv−α1v+β1u, wt=Δw−α2w+β2u in a smooth bounded convex domain Ω⊂R3, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ, ξ, αi and βi(i=1,2) are positive parameters and the logistic source function f fulfills f(s)=s−μsγ+1,s≥0,μ>0andγ≥1. It is shown that this system possesses a unique global bounded classical solution under the conditions αi≥12 and μ≥max⁡{(412χβ1+9ξβ2)γ,(9χβ1+412ξβ2)γ}. Furthermore, whenever u0≢0 and for any γ∈N, the solution of the system approaches to the steady state ((1μ)1γ,(1μ)1γβ1α1,(1μ)1γβ2α2) in the norm of L∞(Ω) as t→∞.