Linear dynamics of semigroups generated by differential operators

Abstract

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on in nite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at in nity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial di erential equations. Additionally, we will also visit the chaos for in nite systems of ordinary di erential equations, that can be of interest for representing birth-and-death process or car-following tra c models