• Evolution equations describe how a state evolves over time in a dynamical system, often governed by differential equations. These equations are fundamental in various fields, including physics, biology, and finance.
• This course provides an in-depth study of C₀-semigroups and their applications to evolution equations, primarily focusing on the theory and methods used to analyze linear partial differential equations (PDEs) and dynamical systems.
• The course covers fundamental concepts, including the generation of semigroups, properties of semigroups, and applications to various types of evolution equations such as parabolic and hyperbolic equations. Emphasis is placed on both theoretical aspects and practical applications.
• Prerequisites: Functional Analysis, Partial Differential Equations, Basic knowledge of Banach and Hilbert spaces.

• Introduction to Semigroups : Definition and examples, Strong continuity, infinitesimal generators,
• Generation results : Hille-Yosida Theorem, Lumer-Phillips Theorem, Examples
• Properties of C₀-Semigroups : Exponential boundedness, Compact semigroups, Differentiability and analyticity of semigroups.
• Spectral properties and asymptotic beahviours
• Perturbation of generators : Kato-Voigt perturbation theorem
• Evolution Equations : Abstract Cauchy problems, Inhomogeneous evolution equations, Well-posedness and regularity of solutions.
• Parabolic Equations : Heat equation as a semigroup, Maximum principles, Long-term behavior of solutions
• Hyperbolic Equations : Wave equation and semigroups, Energy methods, D'Alembert's formula and semigroup approach.

Course slides and reading material will be available from the instructor.

Optional Textbooks:

• Engel, K.-J., and Nagel, R. : One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag 2000.
• Pazy, A. : Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag 1983.
• Renardy, M., and Rogers, R. C. : An Introduction to Partial Differential Equations. Springer-Verlag 2004.

1. Midterm Exam : 25%
2. Project : 25%
3. Final Exam : 50%