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

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system {u(t) = Delta u - del (.) (chi u del v) + del (.) (xi u del w), x is an element of Omega, t > 0, v(t) = D(1)v + alpha u - beta v, x is an element of Omega, t > 0, ((*)) w(t) = D-2 Delta w + gamma u - delta w, x is an element of Omega, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), w(x, 0) = w(0)(x) x is an element of Omega, in a bounded domain It Omega subset of 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)) is an element of [W-1, (infinity) (Omega)](3). Precisely, we show that if the parameters satisfy xi gamma/chi alpha >= max { D-1/D-2, D-2/D-1, beta/delta, delta/beta} for all positive parameters D-1, D-2, chi, xi, alpha, beta, gamma, and delta, the system ((*)) has a unique global classical solution (u, v, w), which converges to the constant steady state ((u) over bar (0), alpha/beta(u) over bar (0), gamma/delta (u) over bar (0)) as t -> +infinity, where (u) over bar (0) = 1/vertical bar Omega vertical bar integral(Omega) u(0)dx. Furthermore, the decay rate is exponential if xi gamma/chi alpha > max {beta/delta, delta/beta}. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e. D-1 not equal D-2) in multi-dimensions.