Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source

This paper is devoted to studying the following quasilinear parabolic-elliptic -elliptic chemotaxis system {u(t) = del . (phi(u)del u - psi(u)del v) + au - bu(gamma), x is an element of Omega, t > 0, 0 = Delta v - v + w(gamma 1), x is an element of Omega, t > 0, 0 = Delta w - w + u(gamma 2), x is an element of Omega, t > 0, with homogeneous Neumann boundary conditions in a bounded and smooth domain Omega subset of R-n(n >= 1), where a, b, gamma(2) > 0, gamma(1) >= 1, gamma > 1 and the functions phi, psi is an element of C-2([0, infinity) satisfy phi(s) >= a(0)(s + 1)(alpha) and vertical bar psi(s)vertical bar <= b(0)s(1 + s)(beta-1) for all s >= 0 with a(0),b(0) > 0 and alpha, beta is an element of R. It is proved that if gamma - beta >= gamma(1)gamma(2), the classical solution of system would be globally bounded. Furthermore, a specific model for gamma(1) = 1, gamma(2) = kappa and gamma = kappa + 1 with kappa > 0 is considered. If beta <= 1 and b > 0 is large enough, there exist C-kappa, mu(1), mu(2) > 0 such that the solution(u, v, w) satisfies parallel to u(., t) - (b/a)(1/kappa)parallel to(L infinity(Omega)) + parallel to v(., t) - b/a parallel to(L infinity(Omega)) + parallel to w(., t) - b/a parallel to(L infinity(Omega)) <= {C(kappa)e(-mu 1t), if kappa is an element of (0, 1], C kappa e(-mu 2t), if kappa is an element of (1, infinity), for all t >= 0. The above results generalize some existing results.