Notes on spectrum and exponential decay in nonautonomous evolutionary equations

Abstract

Abstract. We first determine the dichotomy (Sacker-Sell) spectrum for certain nonautonomous linear evolutionary equations induced by a class of parabolic PDE systems. Having this information at hand, we underline the applicability of our second result: If the widths of the gaps in the dichotomy spectrum are bounded away from 0, then one can rule out the existence of super-exponentially decaying (i.e. slow) solutions of semi-linear evolutionary equations.