Large time behavior of solutions for the attraction-repulsion Keller-Segel system with large initial data

In this paper, we study the following attraction-repulsion Keller-Segel system ut=Δu−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,in a bounded domain Ω⊂R2 with smooth boundary. The boundedness of solutions with arbitrarily large initial data has been proved in the case of ξγ≥χα (Jin and Wang, 2016). Under the additional assumption ξγβ≥χαδ, we show that the global classical solution will converge to the unique constant state (ū0,αβū0,γδū0) as t→+∞.