Global stabilization of the full attraction-repulsion Keller-Segel system

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system{document}$\{ {{array}{*{20}{l}}{{u_t} = Δ u - ∇ · (χ u∇ v) + ∇ · (ξ u∇ w),}&{x ∈ Ω ,t > 0,}\\{{v_t} = {D_1}Δ v + α u - β v,}&{x ∈ Ω ,t > 0,}\\{{w_t} = {D_2}Δ w + γ u - δ w,}&{x ∈ Ω ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x ∈ Ω ,}{array}} .\;\;\;\;( * )documentin a bounded domain Ω⊂ R^2 $ with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (*) with large initial data (u₀,v₀,w₀) ∈ [W1,∞(Ω)]³. Precisely, we show that if the parameters satisfy ξγ/χα≥ max₁/D₂,D₂/D₁,β/δ,δ/β\ for all positive parameters D₁,D₂,χ,ξ,α,β,γ and δ, the system (*) has a unique global classical solution $ (u,v,w) $, which converges to the constant steady state (ū₀,α/̄₀,γ/̄₀) as t→+∞, where ū₀ = 1/|Ω|∫_Ω u₀dx. Furthermore, the decay rate is exponential if ξγ/χα> max\β/δ,δ/β\. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D₁≠ D₂) in multi-dimensions.