Global Stabilization of the Full Attraction-repulsion Keller-Segel System

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system x ∈ Ω, t > 0, x ∈ Ω, t > 0_u w ^{u vt} ₍ ^t t _x ^{= =} = ₀₎ ^{D ∆} D ^{1 u} _{= 2} ^∆ ∆ ⁻ _u ^vw₀ ^{∇ +} ₍+ _x ^{· αu} ₎ ⁽γu ^χu _v ⁻ ₍− ^∇ _x ^{βv, v} δw, ₀₎ ^{) +} ₌ ^∇ _{v 0} ^· ₍ ⁽ _x ^ξu ₎ ^∇ _w ^w _(x ⁾ ₀₎ = w0(x) x ∈ _Ω (∗) x ∈ Ω, t > 0, in a bounded domain Ω ⊂ R² 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) ∈ [W^1,∞(Ω)]³. Precisely, we show that if the parameters satisfy _χα ^ξγ ≥ max ^{n D} _D ¹ _{2 D 1} , δ, _β ^{δ o} for D₂ β 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 _β ū₀ ^γ _δ ū₀) as t → +∞, where ū₀ = _{| Ω} ¹ _| ^R _Ω u₀dx. Furthermore, the decay rate is exponential if _χα ^ξγ > max ^{n β} _{δ β} ^{δ o} . This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D₁ 6= D₂) in multi-dimensions.