# 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.

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.

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 evolution equations, treated by methods of differential equations in Banach spaces (semigroup theory).

The course gives an overview of evolution equations, treated by methods of differential equations in Banach spaces (semigroup theory).

## Course contents

Banach space valued functions of one real variable.

Linear 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 homogenoeus Cauchy problems. Applications to Cauchy problems for linear evolutionary PDEs.

Sectorial operators and analytic semigroups.

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

Banach space valued functions of one real variable.

Linear 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 homogenoeus Cauchy problems. Applications to Cauchy problems for linear evolutionary PDEs.

Sectorial operators and analytic semigroups.

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

## Recommended readings

K. Engel, R. Nagel: One parameter semigroups for linear evolution equations, Springer-Verlag, Berlin, 2000

Lecture notes written by the teacher.

K. Engel, R. Nagel: One parameter semigroups for linear evolution equations, Springer-Verlag, Berlin, 2000

Lecture notes written by the teacher.

## Teaching methods

Lectures in the classroom. Exercises will be assigned, that should be solved and illustrated by the students.

Lectures in the classroom. Exercises will be assigned, that should be solved and illustrated by the students.

## 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.

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.