On the Long-Term Behaviour of Operator Semigroups

In this thesis, we investigate the long-term behaviour of operator semigroups on Banach spaces, with a particular emphasis on positive semigroups on Banach lattices. The main body of this thesis is organized into two distinct parts. Part I focuses on investigating the long-term behaviour of semigroups that lack strong continuity and may have weaker or even no time regularity properties. In Chapter 1, the concept of the semigroup at infinity is used alongside the Jacobs–de Leeuw–Glicksberg decomposition to prove convergence results for abstract semi-group representations on Banach spaces with respect to the operator norm. In Chapter 2, the framework of semigroups on norming dual pairs is used. In this context, a novel monotone approximation theorem for strong Feller semigroups is proven. In Chapter 3, methods from the first chapter are drawn upon and results from classical Perron-Frobenius theory are used to establish an abstract result regarding the long-term behaviour of positive semigroups containing a multiplication operator.

Part II of this thesis delves into the long-term behaviour of classical $C_0$-semigroups. Chapter 4 discusses the asymptotic behaviour of solutions to parabolic PDEs, subject to Neumann boundary conditions, on a domain which are coupled by a potential. In Chapter 5, various systems from queueing and reliability theory are treated within one general, theoretical framework. It is shown that the well-posedness of the equations describing these systems is a consequence of a single generator theorem. Moreover, a new approach to the long-term behaviour of such systems is investigated. In Chapter 6, a coupled transport equation from mathematical biology is discussed. Using a recent result on the norm convergence of $C_0$-semigroups, it is proven that the solutions of said equation converge uniformly to equilibrium.

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