Summer 2012

Seminar on Blow-up in Dynamical Systems

Prof. Dr. Bernold Fiedler, Dr. Stefan Liebscher, Hannes Stuke

July 14-18, 2012, Preliminary discussion will be at the 08.05.12 after the lecture

Goal

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 u_t = Δu + u^p and u_t = Δu + |u|^p-1u.

References

• 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 u_t = Δu + u^{1 + α}. 1966

Target audience

Students of semesters 6-10, students of the BMS (talks can be given in German and/or English)

Prerequisites

Experience in Dynamical Systems or Partial Differential Equations

Topics

Depending on prior knowledge and interests of the participants, talks will cover a selection of the following topics.

PDE basics

• References: Evans; Gilbarg-Trudinger; Jost; or other books on PDE
• Scope: Maximum principle; super- and subsolutions; Sobolev spaces; weak solutions

Local existence results

• References: Pazy
• Scope: Well-posedness; classical, distributional, mild, weak solutions; analytic semigroups; maximal time of existence

Simple blow-up [1-2 Talks]

• References: Hu, Chapter 5.1-5.3; Quittner / Souplet, Chapter II.17
• Scope: Kaplan eigenvalue method, Concavity method, Comparison principle, Starting above positive equilibrium

Critical exponent

• References: Quittner / Souplet, Chapter II.18; Fujita
• Scope: Fujita type results, Dependence of the dimension

Diffusion vs Blow-up

• References: Quittner / Souplet, Chapter II.19.3
• Scope: Diffusion eliminates blow-up, Discussion of examples

Blow-up set

• References: Quittner / Souplet, Chapter II.24; Hu, Chapter 7
• Scope: What is a blow-up set?, How does it look like --- discrete, compact?

Blow-up rate

• References: Hu, Chapter 7; Quittner / Souplet Chapter II.23
• Scope: 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

• References: Hu, Chapter 8; Giga / Kohn
• Scope: Similarity variables, Backward self-similarity, Asymptotics of the blow-up solutions, Stationary solutions and blow-up

Shape of blow-up: center-manifold approach

• References: Fila / Matano; Fillipas
• Scope: Center manifolds, slaving principle, Center dynamics and shapes

Shape of blow-up: Prescribed shape in unstable manifolds

• References: Fiedler / Matano
• Scope: Understanding the paper

Beyond blow-up

• References: Fila / Matano; Quittner / Souplet, Chapter II.27
• Scope: Complete vs incomplete blow-up, Continuation of blow-up solutions