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 . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense 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 χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.