### Nonlinear Evolution Equations (Summer Semester 2024)

- Lecturer: Prof. Dr. Roland Schnaubelt
- Classes: Lecture (0156500), Problem class (0156510)
- Weekly hours: 4+2
- Audience: Mathematics (from 8. semester)

**All information and materials for this lecture are provided in the course "Evolution Equations" within Ilias, including communication via email and forum. Please join this course if you want to participate.**

Schedule | |||
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Lecture: | Monday 14:00-15:30 | SR 2.066 | Begin: 15.4.2024 |

Wednesday 11:30-13:00 | SR 2.059 | ||

Problem class: | Friday 11:30-13:00 | SR 0.019 | Begin: 19.4.2024 |

Lecturers | ||
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Lecturer | Prof. Dr. Roland Schnaubelt | |

Office hours: Tuesday at 12:00 - 13:00, and by appointment. | ||

Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30) | ||

Email: schnaubelt@kit.edu | ||

Problem classes | Richard Nutt M.Sc. | |

Office hours: Just drop by whenever I am in the office. | ||

Room 2.043 Kollegiengebäude Mathematik (20.30) | ||

Email: richard.nutt@kit.edu |

Evolution equations describe the time evolution of dynamical systems by an ordinary differential equation in a Banach or Hilbert space. In this lecture we study nonlinear and autonomous (time-invariant) problems whose main part is given by the generator of a linear, strongly continuous operator semigroup. In particular, we deal with reaction-diffusion equations and the semilinear wave and Schrödinger equations.

Typical questions are existence and uniqueness, continuous dependence on the data, blow-up versus global-in-time existence, regularity, or the long-term behavior in the vicinity of equilibria. Many of the statements and methods are based on the theory of ordinary differential equations (Analysis 4), even if the presence of discontinuous operators in Banach spaces leads to numerous new and profound difficulties and phenomena. Our approach is essentially based on functional analytical thinking and results.

It is strongly recommended to have attended the lectures Functional Analysis and Evolution Equations. However, the necessary contents of the latter lecture will be briefly repeated.

# Examination

The module examination will be oral (duration approx. 30 min). Details will be communicated later here in the lecture, here and on Ilias.

# References

On my homepage and in Ilias lecture notes are provided in sections as PDF. Here are some relevant monographs.

- T. Cazenave: Semilinear Schrödinger Equations. American Math. Soc., 2003.
- T. Cazenave, A. Haraux: An Introduction to Semilinear Evolution Equations.Oxford Univ. Press, 1998.
- D. Henry: Geometric Theory of Semilinear Parabolic Equations. Springer, 1981.
- A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, 1995.
- A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, 1983.
- C. Sogge: Lectures on Non-linear Wave Equations. International Press, 2008.
- T. Tao: Nonlinear Dispersive Equations. American Math. Soc., 2006.
- R. Temam: Infinite-dimensional Dynamical Systems in Mechanics and Physics. Springer, 1997.