SoSe2026
SoSe2026
Time & Place
Lectures: Thursday, 11:00-13:00 (weekly)
Exercises: Thursday, 13:00-15:00 (every two weeks)
Short description
In this course, we study the theory of abstract evolution equations, which provides a unified mathematical framework for describing how systems evolve over time. Such equations arise in many areas of science and engineering, for example in models of diffusion (heat flow), wave propagation, and other dynamical processes. The course introduces operator semigroup theory as a powerful and conceptually simple tool for analysing these problems.
We begin with the autonomous case, where the governing equations do not depend explicitly on time. In this setting, we develop the theory of $C_0$-semigroups (strongly continuous one-parameter semigroups), which provide a natural extension of the classical exponential solution formula for linear differential equations to infinite-dimensional spaces. A central result is the Hille-Yosida theorem, which characterizes when a linear operator generates a semigroup and thus determines a well-posed evolution equation. We will also see how to construct semigroups for concrete classes of differential equations.
We then turn to nonautonomous evolution equations, where coefficients vary in time. These require a more general framework based on evolution families (two-parameter operator families), which play a key role in understanding well-posedness in the time-dependent setting.
Finally, we explore how semigroup methods can be used to study qualitative properties of solutions, such as stability and long-time behavior, and how they can be applied to treat perturbations and inhomogeneous problems.
Prerequisites
Basic knowledge of Functional Analysis and (Partial) Differential Equations.
References
K. Engel, R.Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, 2000.
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.
K. Yosida. Functional Analysis, Springer, 1980.