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

Abstract

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system ut = ∆u − ∇ · (χu∇v) + ∇ · (ξu∇w), x ∈ Ω, t > 0, vt = D1∆v + αu − βv, x ∈ Ω, t > 0, wt = D2∆w + γu − δw, x ∈ Ω, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), w(x, 0) = w0(x) x ∈ Ω, (∗) 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) ∈ [W1,∞(Ω)]3 . Precisely, we show that if the parameters satisfy ξγ χα ≥ max n D1 D2 , D2 D1 , β δ , δ β o 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 (¯u0, α β u¯0, γ δ u¯0) as t → +∞, where ¯u0 = 1 |Ω| R Ω u0dx. Furthermore, the decay rate is exponential if ξγ χα > max n β δ , δ β o . This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D1 =/= D2) in multi-dimensions. [Abstract not complete, refer to publisher pdf]