Perturbation Methods for Abstract Cauchy Problems associated to Evolution Equations

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In this project we study the perturbation theory for abstract Cauchy problems associated to evolution equations. Many well-known partial differential equations modeling physical systems, such as the heat equation, the Schrödinger equation or the wave equation, use temporal change of states. "Evolution equation" is an umbrella term for such equations that can be interpreted as differential laws describing the development of a system or as a mathematical treatment of motion in time. By modeling systems evolving in time, the time variable plays a crucial role, as the equations are built by balancing the change of the system in time against its "spatial" behaviour. By finding a solution of an evolution equation one has the possibility to predict the future of the corresponding physical system which makes it deterministic. In an ideal world, one would be able to find an exact solution in terms of elementary functions for a given equation since the evolution of the system is given explicitly. This would be beneficial for practitioners. However, most of the really interesting equations cannot be treated in such a way. Nonetheless, the study of existence of solutions, their uniqueness, as well as the qualitative properties serve as a partial validation of the used models and also provides a foundation for the numerical analysis of the equation which is important for the practitioners. Perturbation theory is a powerful tool in this regard and allows for a more general and abstract view on such problems. The general idea is to start with a related and simpler problem where the exact solution is already known.On the one hand side, we study autonomous dynamical systems. We will not only consider C_0-semigroups but also non-strongly continuous semigroups or semigroups on vector lattices by means of bi-continuous semigroups and ru-continuous semigroups, respectively. On the other hand side, we also consider time-dependent and hence non-autonomous dynamical systems of different orders.