Given a square matrix A, the first order system d/dt u(t) = A u(t) can be solved by the matrix exponential: u(t) = exp(t A) u(0). In particular, as a function of t this gives rise to a one-parameter group.

Similarly, partial differential equations (e.g., the heat equation) fit within this framework if the matrix A is replaced by a suitable operator. For the heat equation, it is the Laplace operator. However, since the Laplace operator is not bounded, the series for the exponential will not converge and this raises the question how the exponential should be defined in this case. Moreover, solutions of the heat equation exist in general only for positive times and the solution operator can form at best a semigroup.

This course aims at developing a solid mathematical foundation to this approach. As a byproduct, we obtain a very general solution concept for certain evolution equations and a framework to characterize long-term behavior of the solutions.

### Organization

Lecture: Thu 8:30am - 10:05am (Room: A3/115)

Exercise: Wed 2:15pm - 3:45pm (Room: A6/017)

#### Contact

Péter Koltai (peter.koltai@fu-berlin.de)

#### Exercise class

There will be exercise sheets. It is expected that you prepare presentations of solutions to the exercises - you will not* *be required to submit written solutions. However, for the **active participation **("aktive Teilnahme") you will be required to **present at least four solutions**, at least two in each half of the semester.

**There will be no exercise class on April 20. The first event in this course is the lecture on Thu, April 21.**

Sheets and additional material will be provided in Resources.

#### Exams

Information will be provided in due course.

### Prerequisites

- Calculus (Analysis I-III)
- Linear algebra (Lineare Algebra I-II)
- Functional analysis

### Literature

Operator semigroups:

- K.-J. Engel and R. Nagel.
*One-parameter semigroups for linear evolution equations*. Springer. 2000. - K.-J. Engel and R. Nagel.
*A short course on operator semigroups*. Springer. 2006. - A. Pazy.
*Semigroups of linear operators and applications to partial differential equations*. Springer. 1983. - D. Hundertmark, L. Machinek, M. Meyries, and R. Schnaubelt.
*Operator semigroups and dispersive equations*. 16th Internet Seminar of Evolution Equations. 2013.

Functional analysis:

- [Con] J. B. Conway,
*A course in functional analysis*, 2nd edition, Springer, 1990. - [Kre] E. Kreyszig.
*Introductory Functional Analysis with Applications*. Wiley, 1978. - [Rud] W. Rudin,
*Functional Analysis*, McGraw-Hill, 1991. - [Wer] D. Werner,
*Funktionalanalysis*, 7. Aufl., Springer, 2011.