Reduction principle for partial functional differential equation without compactness

This article establishes a reduction principle for partial functional differential equation without compactness of the semigroup generated by the linear part. Under conditions more general than the compactness of the C0-semigroup generated by the linear part, we establish the quasi-compactness of the C0-semigroup associated to the linear part of the partial functional differential equation. This result allows as to construct a reduced system that is posed by an ordinary differential equation posed in a finite dimensional space. Through this result we study the existence of almost automorphic and almost periodic solutions for partial functional differential equations. For illustration, we study a transport model.