Analytic Semigroups and Parabolic Evolution Equations (V5B6) 

• Thursday 14-16, starting from April 23 
• Lecture Notes 

 

Organisation of the lecture:

If you want to attend this course, held online via zoom, then please send me an e-mail with the following information:

• Your full name
• Your e-mail adress
• Your matriculation number

After you have sent this e-mail I will give you the password needed for the registration on eCampus. There you find the information you need to join the zoom meeting.

 

Content of the lecture:

Important questions in the study of PDEs are those for existence, uniqueness and qualitative behaviour of solutions. In this lecture the focus is on parabolic PDEs and the theory of analytic semigroups as a tool to obtain answers for these questions. The main idea of semigroup theory in this context is to interpret a given PDE as an abstract ODE in an infinite dimensional Banach space. We will see that solutions of linear evolution equations are constructed by a semigroup which may be seen as a generalisation of the matrix exponential for unbounded operators on Banach spaces.

The aim of this lecture is to give an insight into parts of the basic theory of analytic semigroups and how this theory may be used to solve parabolic evolution equations. We first introduce some fundamental results on analytic semigroups and their infinitesimal generators. Then the lecture is mainly concerned with applications of this theory in the qualitative analysis of parabolic PDEs.

 

Prerequisites:

Basic notions of functional analysis and PDEs should be known. Knowledge on strongly continuous semigroups and hyperbolic PDEs is helpful but not necessary.