Robustness of asymptotic properties of evolution families under perturbations

Abstract

In this paper, we present some conditions under which asymptotic properties of an evolution family ${{\mathcal{U}}}:={(U(t,s))_{t\geq s\geq 0}}$ persist under perturbations by a family $\mathcal B:=(B(t),D(B(t))_{t\in{{\mathbb{R_+}}}}$ of linear operators on a Banach space $X$ satisfying suitable conditions. Our results concern asymptotic properties like boundedness, periodicity, and asymptotic almost periodicity (even in the sense of Eberlein). An application of the abstract results to nonautonomous partial differential equations with delay is given.