GLOBAL STABILIZATION OF THE FULL ATTRACTION-REPULSION KELLER-SEGEL SYSTEM

摘要

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system in a bounded domain Ω. C 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 (u_0,v_0,w_0) ∈ [W~(1,∞) (Ω)]~3. Precisely, we show that if the parameters satisfy (ζγ/xα) ≥max {D_1/D_2,D_1/D_2, β/δ, δ/β}for all positive parameters D_1, D_2, X>α, β,γ and δ, the system (*) has a unique global classical solution (u,v,w), which converges to the constant steady state (ū_0, α/βū_0, γ/δū_0) as t → +∞, where ū_0 = 1/|Ω| ∫_Ω u0dx. Furthermore, the decay rate is exponential if ξγ/xα > max { β/δ, δ/β}This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D_1≠D_2) in multi-dimensions.