Linear dynamics of semigroups generated by differential operators

[EN] During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-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 infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.The first, third and fourth authors were supported by MINECO and FEDER, grant MTM2016-75963-P. The second author has been partially supported by CONICYT under FONDECYT grant number 1140258 and CONICYT-PIA-Anillo ACT1416.Conejero, JA.; Lizama, C.; Murillo Arcila, M.; Peris Manguillot, A. (2017). Linear dynamics of semigroups generated by differential operators. Open Mathematics. 15(1):745-767. https://doi.org/10.1515/math-2017-0065745767151Herzog, G. (1997). On a Universality of the Heat Equation. Mathematische Nachrichten, 188(1), 169-171. doi:10.1002/mana.19971880110Kalmes, T. (2010). Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators. 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