Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source

This paper deals with an attraction repulsion chemotaxis system with logistic source {u(t) = Delta u - chi del center dot (u del v) + xi del center dot (u del w) + f(u), x is an element of Omega, t > 0, v(t) = Delta v + alpha u - beta v, x is an element of Omega, t > 0 w(t) = Delta w + gamma u - delta w, x is an element of Omega, t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-N (N >= 1), where parameters chi, xi, alpha, beta, gamma and delta are positive and f(s) = kappa s - mu s(1+k) with kappa is an element of R, mu > 0 and k >= 1. It is shown that the corresponding system possesses a unique global bounded classical solution in the cases k > 1 or k = 1 with mu > C-N mu* for some mu*, C-N > 0. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when kappa < 0 (resp. kappa = 0), the corresponding solution of the system decays to (0,0,0) exponentially (resp. algebraically), and when kappa > 0 the solution converges to ((kappa/mu)(1/k), alpha/beta(kappa/mu)(1/k), gamma/delta(kappa/mu)(1/k)) exponentially if mu is larger. (C) 2017 Elsevier Ltd. All rights reserved.