On a parabolic-elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic-elliptic chemotactic PDEs system, which describes the evolution of a biological population u and a chemical substance v in a bounded domain Omega subset of R-n. We consider a growth term of logistic type in the equation of u in the form mu u(1 - u + f(t,x)). The function f, describing the resources of the systems, presents a periodic asymptotic behavior in the sense lim(t ->infinity) sup(x is an element of Omega)vertical bar f(x, t) - f*(t)vertical bar = 0, where f* is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f*, if the constant chemotactic sensitivity chi satisfies chi < mu/2, we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.