CEU Electronic Theses and Dissertations, 2009
| Author | Sereny, Andras G |
|---|---|
| Title | Some Results on Operator Semigroups and Applications to Evolution Problems |
| Summary | In this thesis we address certain questions arising in the functional analytic study of dynamical systems and differential equations. First, we discuss the operator theoretic counterparts of the central ergodic theoretical notions of strong and weak mixing. These concepts correspond to particular types of asymptotic behaviour of operator semigroups in the weak operator topology, called weak and almost weak stability. Using functional analytic tools and methods from ergodic the- ory, we describe various features of (almost) weakly stable semigroups. In particular, we show that (in the Baire category sense) typical elements in certain natural spaces of semi- groups are almost weakly but not weakly stable, thus we carry over classical theorems of Halmos and Rohlin for measure preserving transformations to the Hilbert space operator setting. Further, we illustrate operator semigroup methods and results on a class of telegraph systems with various boundary conditions. We study both linear and nonlinear boundary value problems. The stability of linear telegraph systems is discussed by applying theorems from the previous chapters. For the existence of solutions, we are particularly interested in time-dependent boundary conditions, since this case has little been investigated so far. The operator semigroup techniques applied to the case of Lipschitz continuous non- linearities are combined with estimates from the theory of monotone operators to yield well-posedness and the regularity of the solutions, also in the case of dynamic boundary conditions. |
| Supervisor | Morosanu, Gheorghe |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2009/tphsea01.pdf |
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