On a fully parabolic chemotaxis system with source term and periodic asymptotic behavior

We study a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population u and a chemical substance v in a two-dimensional bounded domain with regular boundary. We consider a growth term of logistic type in the equation of u in the form u(1-u+f(x,t)), for a given bounded function f which tends to a periodic in time function independent of x when t goes to infinity. We study the global existence of solutions and its asymptotic behavior for a range of parameters and initial data.