In this course we will study the elegant theory of abstract evolution equations (with evolving time), which covers, in particular, parabolic and hyperbolic partial differential equations and functional differential equations with delay. The operator semigroup theory provides a natural and surprisingly simple framework for linear (and some nonlinear) evolution equations.
In the autonomous setting, we will deal with the so-called $C_0$-semigroups, also known as strongly continuous one-parameter semigroups. They provide solutions of linear ordinary differential equations with constant coefficients in Banach spaces. Any $C_0$-semigroup is a representation of the semigroup $(\R_+,+)$ on a Banach space that is continuous in the strong operator topology. The well-known Hille-Yosida theorem, which originated the whole theory, provides a description of linear operators on a Banach space generating the $C_0$-semigroups for a given evolution equation.
Evolution families (strongly continuous two-parameter operator semigroups) are a non-autonomous counterpart of operator semigroups in the well-posedness theory for non-autonomous evolution equations. We will construct $C_0$-semigroups and evolution families for particular differential equations.
The concept of evolution operator semigroups serves also as a suitable setting for investigating stability and long-term behavior of solutions as well as for developing perturbation theory and for solving inhomogeneous problems.
Prerequisites
Basic knowledge of Functional Analysis and Linear Partial Differential Equations.
- Kursverantwortliche/r: PD Dr. habil. Irina Kmit