Course: 2024/2025

Advanced methods for nonlinear differential equations

(18781)

Requirements (Subjects that are assumed to be known)

The course is aimed at master's students with a basic knowledge of the theory of Differential Equations and Analysis. It is recommended to have passed the introductory courses:
-Differential Calculus
-Ordinary differential equations
-Partial Differential Equations
-Real Analysis
-Functional Analysis

The course focuses on the development of the theory of nonlinear differential equations with the aim of familiarising the student with important techniques and results in this nonlinear context.
In particular, it is intended that the student understands the intrinsic problems of nonlinear problems and acquires advanced skills in: fixed point theory and bifurcation theory and their applications to differential equations; in the theory of change of scale and self-similar solutions.
Basic skills: CB6, CB7, CB10
General skills: CG4, CG5, CG6
Specific skills: CE2, CE8

Skills and learning outcomes

Description of contents: programme

1. Fixed Point Theory: Contractive applications and Fixed Point Theorems.
2. Bifurcation Theory: Classification of bifurcations. Global bifurcation.
3. Changes of scale and self-similarity: Classification of solutions. Transformation groups.
4. Applications: semilinear and quasilinear elliptic problems; nonlinear eigenvalue problems; periodic and travelling waves.

Learning activities and methodology

1. THEORETICAL-PRACTICAL CLASSES, where the knowledge that the students must acquire is explained and developed. Students will have basic reference texts to facilitate the understanding of the classes and the development of follow up work. The teacher and the students will solve exercises and practical problems, previously suggested by the teacher.
2. TUTORING SESSIONS. Individualized attendance for students with a teacher.
3. STUDENT INDIVIDUAL OR GROUP WORK. Each student's individualized study, understanding of results and proofs, and exercise and problem-solving is fundamental in mathematics, both for learning and for self-evaluation of acquired competencies and skills.

Assessment System

- % end-of-term-examination 50
- % of continuous assessment (assigments, laboratory, practicals...) 50

Calendar of Continuous assessment

Basic Bibliography

- A. Ambrosetti, A. Malchiodi. Nonlinear Analysis and semilinear elliptic problems. Cambridge University Press. 2007
- G. Barenblatt. Scaling, self-similarity, and intermediate asymptotic. Cambridge University Press. 1996
- Lawrence C. Evans. Partial Differential Equations. American Mathematical Society. 1998
- M. S. Berge. Nonlinearity and Functional Analysis. Academic Press. 1977

Additional Bibliography

- David Gilbarg, Neil S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag Berlin Heidelberg.
- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. 2010
- K. Deimilin. Nonlinear Functional Analysis. Dover. 2009
- P. Drábek, J. Milota. Methods on Nonlinear Analysis. Springer. 2013
- Yuri A. Kuznetsov. Elements of Applied bifurcation Theory. Springer. 1998

The course syllabus may change due academic events or other reasons.