Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems. MSC subject classification: 37B25, 37B55, 93D30

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on Rd which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as t → +∞. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.