Global dynamics in a chemotaxis system involving nonlinear indirect signal secretion and logistic source

This paper is concerned with a quasilinear parabolic-parabolic-elliptic chemotaxis system { u(t) = V <middle dot> (phi(u)Vu - psi(u)del v) + au - bu(gamma), x is an element of Omega, t > 0, v(t)= 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,under homogeneous Neumann boundary conditions in a bounded and smooth domain Omega subset of R-n(n >= 1), where a, b, gamma 1, gamma 2 > 0, gamma > 1, phi and psi are nonlinear functions satisfying phi(s) >= a0(s + 1)alpha and |psi(s)| < b0s(1 + s)beta-1 for all s >= 0 with a(0), b(0) > 0 and alpha, beta is an element of R. When beta + gamma 1 gamma 2 < max{ (n+2)/(n) + alpha, gamma}, then the system has a classical solution which is globally bounded in time. Moreover, when beta + gamma 1 gamma 2 = max{ (n+2)/(n) + alpha, gamma}, it has been shown that the existence of global bounded classical solution depends on the size of coefficient b and initial data u0. Furthermore, we consider a specific system with gamma 1 = 1, gamma 2 = kappa and gamma = kappa + 1 for kappa > 0. If b > 0 is sufficiently large, the global classical solution(u, v, w) exponentially converges to the steady state ((b/a) (1/kappa) ,a/b,b/a ) in L-infinity norm as t -> (infinity), where convergence rate is explicitly expressed in terms of the system parameters.