Summer 2012
Seminar on Blowup in Dynamical Systems
Prof. Dr. Bernold Fiedler,
Dr. Stefan Liebscher,
Hannes Stuke
July 1418, 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 “blowup”?
 Are there different types? Can we categorize them?
 How does the blowup look like? Can we quantify it?
 Can we continue solutions after the blowup?
Central examples will be reactiondiffusion equations of the types
u_{t} = Δu + u^{p}
and
u_{t} = Δu + u^{p1}u.
References
 A. Pazy:
Semigroups of Linear Operators and Applications to Partia Differential Equations. Springer, 1983
 P. Quittner / P. Souplet:
Superlinear Parabolic Problems  Blowup, Global Existence and Steady States. Springer Basel, 2007
 B. Hu:
Blowup Theories for Semilinear Parabolic Equations. Springer, 2011
 M. Fila / H. Matano:
Blowup in nonlinear heat equations from the dynamical systems point of view. Handbook of Dynamical Systems 2, Chapter 14, 2002
 Y. Giga / R. Kohn:
Asymptotically Selfsimilar Blowup of Semilinear Heat Equations. Communications on Pure and Applied Mathematics, 1985
 S. Fillipas:
On the blowup of multidimensional semilinear heat equations. IMA Preprint Series 798, 1991
 B. Fiedler / H. Matano:
Global Dynamics of Blowup Profiles in Onedimensional 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 610, 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; GilbargTrudinger; Jost; or other books on PDE
 Scope:
Maximum principle; super and subsolutions; Sobolev spaces; weak solutions
Local existence results
 References:
Pazy
 Scope:
Wellposedness; classical, distributional, mild, weak solutions;
analytic semigroups; maximal time of existence
Simple blowup [12 Talks]
 References:
Hu, Chapter 5.15.3; 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 Blowup
 References:
Quittner / Souplet, Chapter II.19.3
 Scope:
Diffusion eliminates blowup, Discussion of examples
Blowup set
 References:
Quittner / Souplet, Chapter II.24; Hu, Chapter 7
 Scope:
What is a blowup set?, How does it look like  discrete, compact?
Blowup rate
 References:
Hu, Chapter 7; Quittner / Souplet Chapter II.23
 Scope:
Lower and upper bound of the blowup rate based on scaling methods, Blowup rates for examples
Shape of blowup via energy estimates
 References:
Hu, Chapter 8; Giga / Kohn
 Scope:
Similarity variables, Backward selfsimilarity, Asymptotics of the blowup solutions, Stationary solutions and blowup
Shape of blowup: centermanifold approach
 References:
Fila / Matano; Fillipas
 Scope:
Center manifolds, slaving principle, Center dynamics and shapes
Shape of blowup: Prescribed shape in unstable manifolds
 References:
Fiedler / Matano
 Scope:
Understanding the paper
Beyond blowup
 References:
Fila / Matano; Quittner / Souplet, Chapter II.27
 Scope:
Complete vs incomplete blowup, Continuation of blowup solutions
