Seminar on Blow-up in Dynamical Systems
Prof. Dr. Bernold Fiedler,
Dr. Stefan Liebscher,
July 14-18, 2012, Preliminary discussion will be at the 08.05.12 after the lecture
Recent PDE research was mainly dominated by the question of existence and regularity of solutions of certain PDEs. But in applications, like physics, there naturally arise PDEs, which possess solutions that do not stay globally bounded. In this seminar we want to study verious aspects of solutions that “blow up”, that is to say, stop to exists in a certain sense.
We want to adress questions like
- What is the mathematical framework to define “blow-up”?
- Are there different types? Can we categorize them?
- How does the blow-up look like? Can we quantify it?
- Can we continue solutions after the blow-up?
Central examples will be reaction-diffusion equations of the types
ut = Δu + up
ut = Δu + |u|p-1u.
- A. Pazy:
Semigroups of Linear Operators and Applications to Partia Differential Equations. Springer, 1983
- P. Quittner / P. Souplet:
Superlinear Parabolic Problems - Blow-up, Global Existence and Steady States. Springer Basel, 2007
- B. Hu:
Blow-up Theories for Semilinear Parabolic Equations. Springer, 2011
- M. Fila / H. Matano:
Blow-up in nonlinear heat equations from the dynamical systems point of view. Handbook of Dynamical Systems 2, Chapter 14, 2002
- Y. Giga / R. Kohn:
Asymptotically Self-similar Blow-up of Semilinear Heat Equations. Communications on Pure and Applied Mathematics, 1985
- S. Fillipas:
On the blow-up of multidimensional semilinear heat equations. IMA Preprint Series 798, 1991
- B. Fiedler / H. Matano:
Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations. Journal of Dynamics and Differential Equations, 2007
- H. Fujita:
On the blowing up of solutions of the Cauchy problem for ut = Δu + u1 + α. 1966
Students of semesters 6-10, students of the BMS (talks can be given in German and/or English)
Experience in Dynamical Systems or Partial Differential Equations
Depending on prior knowledge and interests of the participants,
talks will cover a selection of the following topics.
Evans; Gilbarg-Trudinger; Jost; or other books on PDE
Maximum principle; super- and subsolutions; Sobolev spaces; weak solutions
Local existence results
Well-posedness; classical, distributional, mild, weak solutions;
analytic semigroups; maximal time of existence
Simple blow-up [1-2 Talks]
Hu, Chapter 5.1-5.3; Quittner / Souplet, Chapter II.17
Kaplan eigenvalue method, Concavity method, Comparison principle, Starting above positive equilibrium
Quittner / Souplet, Chapter II.18; Fujita
Fujita type results, Dependence of the dimension
Diffusion vs Blow-up
Quittner / Souplet, Chapter II.19.3
Diffusion eliminates blow-up, Discussion of examples
Quittner / Souplet, Chapter II.24; Hu, Chapter 7
What is a blow-up set?, How does it look like --- discrete, compact?
Hu, Chapter 7; Quittner / Souplet Chapter II.23
Lower and upper bound of the blow-up rate based on scaling methods, Blow-up rates for examples
Shape of blow-up via energy estimates
Hu, Chapter 8; Giga / Kohn
Similarity variables, Backward self-similarity, Asymptotics of the blow-up solutions, Stationary solutions and blow-up
Shape of blow-up: center-manifold approach
Fila / Matano; Fillipas
Center manifolds, slaving principle, Center dynamics and shapes
Shape of blow-up: Prescribed shape in unstable manifolds
Fiedler / Matano
Understanding the paper
Fila / Matano; Quittner / Souplet, Chapter II.27
Complete vs incomplete blow-up, Continuation of blow-up solutions