# EVOLUTION EQUATIONS

## Learning outcomes of the course unit

At the end of the course students should know the basic theory of semigroups of operators n Banach spaces and how to apply such theory to different types of evolution PDEs.

Through exercises solved in the classroom students should understand how to apply his/her theoretic knowledges to solve explicit problems.

Students should be able to evaluate the correctness of the results obtained by himself/herself or by other people.

Students should be able to communicate in a clear and precise way the mathematical contents of the course. Lectures in the classroom and discussion with the teacher will help to be able to use the appropriate scientific language.

Students will be able to deepen their knowledge on the subjects of the course, starting from the basic knowledge given by the course itself. They will be able to consult autonomously specialized monographs, even on related subjects not directly treated in the lectures.

## Prerequisites

Calculus for functions of several variables. Linear algebra. Topology. Lebesgue measure theory and integration.

Basic theory of linear functional analysis.

The knowledge of Sobolev spaces with respect to the Lebesgue measure is recommended, although not essential.

## Course contents summary

The course gives an overview of linear evolution equations, treated by methods of differential equations in Banach spaces (semigroup theory). To this aim, some tools that are not considered in the other courses in Functional Analysis of our university are provided, such as spectral theory of linear operators in Banach spaces and integrals of real or complex variable functions with values in Banach spaces. Such basic tools are used to develop the Semigroup theory, both strongly continuous and analytic. Both homogeneous and non-homogeneous Cauchy problems are studied, addressing to questions of existence, uniqueness, regularity and asymptotic behavior of the solutions.

## Course contents

Banach space valued functions of one real variable, integrals and derivatives.

Banach space valued functions of one complex variable, contour integrals.

Linear differential equations and Cauchy problems with bounded operators.

Spectrum, resolvent, and spectral properties of linear operators in Banach spaces.

Strongly continuous semigroups and their infinitesimal generators.

The Hille-Yosida Theorem.

Non homogenous Cauchy problems. Applications to Cauchy problems for linear evolutionary PDEs.

Sectorial operators and analytic semigroups.

Regularity of solutions of Cauchy problems, both homogeneous and non-homogeneous.

Asymptotic behavior in homogeneous and in nonhomogeneous problems. Applications to PDEs of parabolic type.

## Recommended readings

Lecture notes written by the teacher;

K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, 2000.

## Teaching methods

Lectures in the classroom. Exercises will be assigned, that should be solved and illustrated by the students. Four hours per week will be devoted to lectures and two hours per week will be devoted to exercises.

If the classrooms will not be available because of the Covid epidemics, lectures will be held online through Microsoft teams.

## Assessment methods and criteria

The examination consists of an oral test which is aimed at evaluating the knowledge of the results seen during the course, their proofs, and the skills in using such results to solve simple problems in the fields of the course.

If classrooms will not be available due to the Covid epidemics, examinations will be online through Microsoft Teams.