Reduction principle and dynamic behaviors for a class of partial functional differential equations

We investigate the dynamic of solutions in the αα-norm for some nonhomogeneous linear partial functional differential equations. We suppose that the undelayed homogeneous part is the infinitesimal generator of an analytic semigroup, and the delayed part is continuous with respect to fractional powers of the generator. We establish a reduction principle for the infinite dimensional system in order to reduce its qualitative analysis to a finite dimensional one. Our reduction method is based on a new variation of constants formula. In application, the reduced system is used to prove the existence of almost automorphic, almost periodic and periodic solutions for the whole infinite dimensional system.