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.