BOUNDEDNESS AND LARGE TIME BEHAVIOR OF AN ATTRACTION-REPULSION CHEMOTAXIS MODEL WITH LOGISTIC SOURCE

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source {u(t) - Delta u - chi Delta . (u Delta v) + xi Delta . (u del w) + f(u), x epsilon Omega, t > 0, u(t) = Delta v + alpha u -beta v, x epsilon Omega, t > 0, w(t) = Delta w + gamma u - delta w, x epsilon Omega, t > 0, in a smooth bounded domain Omega subset of R-n (n >= 1), with homogeneous Neumann boundary conditions and nonnegative initial data (uo, vo, wo) satisfying suitable regularity, where chi >= 0, xi >= 0, alpha, beta, gamma, delta > 0 and f is a smooth growth source satisfying f (0) >= 0 and f (u) <= a bu(0), u >= 0, with some a >= 0, b > 0, theta >= 1 When chi alpha = xi gamma (i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter theta and the space dimension n satisfy { theta > max (1, 3 - 6/n), when 1 <= n <= 5, theta <= 2, when 6 <= n <= 9, theta > 1 + 2(n-4)/n+2, when n >= 10. Furthermore, when f(u) = mu u(1 u) and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if it mu > chi(2) alpha(2) (beta-delta)(2)/8 delta beta(2), the solution (u, v, w) exponentially stabilizes to the constant stationary solution (1, alpha/beta, gamma/delta) in the case of 1 <= n <= 9. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.