Winter semester 2016/17
Seminar: Blowup in differential equations
Prof. Dr. Bernold Fiedler,
Hannes Stuke
Schedule, Winter 2016/17
 Seminar:
 The seminar will take place from 08.10.2016  13.10.2016 in the Sächsische Schweiz
Description
In this seminar we want to address the phenomenon of blowup in ordinary differential and partial differential equations. Blowup means, that solutions become unbounded in finite time, which leads to many interesting questions, e.g.
 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 blowup?
The discussion of the topics will take place on 19.07.2016 at 12.15 in seminar room 140.
Topics
Blowup of quadratic systems
 Existence of blowup
 Asymptotic behaviour
 References: [1]
Blowup criteria and blowup rate estimates
 Concavity method to prove existence of blowup
 Upper bound on blowup rate
 References: [2], [3]
Blowup in rescaled variables
 Selfsimilar variables
 Stationary solutions of rescaled flow
 Convergence to stationary solutions
 References: [2], [3], [4]
Blowup from dynamical systems point of view
 Problem formulation
 Linearization and higher order asymptotics
 References: [5], [6], [7]
Description of the blowup profile
 Description of blowup profile for a single point
 References: [3]
Zero numbers and blowup profile
 Zero numbers
 Construction of blowup profile with n  blowup points
 References: [2], [8]
Complete blowup
 Definition of complete blowup
 Existence of complete blowup
 Existence of noncomplete blowup
 References: [2], [3]
References
 [1] SzeBi Hsu, Bernold Fiedler, HsiuHau Lin. Classification of potential flows under renormamlization group transformation. Discrete Contin. Dyn. Syst., Ser. B 21, pp. 437446, (2016).
 [2] Bei Hu. Blowup Theories for Semilinear Parabolic Equations. Springer, (2011).
 [3] Pavol Quittner, Phillippe Souplet. Superlinear Parabolic Problems
Birkhäuser, (2007).
 [4] Yoshikazu Giga. Robert V. Kohn. Asymptotically Selfsimilar Blowup
of Semilinear Heat Equations. Communications on Pure and Applied Mathematics, Vol. XXXVIII, pp. 297319, (1985).
 [5] Marek Fila, Hiroshi Matano. Blowup in Nonlinear Heat Equations
from the Dynamical Systems Point of View. Hanbook of Dynamical Systems, VOL. 2, pp. 723758, (2002).
 [6] Stathis Filippas. Wenxiong Liu. On the blowup of multidimensional semilinear heat equations. IMA Preprint Series # 798, (1991).
 [7] J. J. L. Velazquez, V. A. Galaktionov, M. A. Herrero. The space structure near a blowup point for semilinear heat equations: A formal approach, Zh. Vychisl. Mat. Mat. Fiz., Volume 31, Number 3, pp. 399–411, (1991).
 [8] Bernold Fiedler, Hiroshi Matano. Global Dynamics of Blowup Profiles
in Onedimensional Reaction Diffusion Equations. Journal of Dynamics and Differential Equations, Vol. 19, No. 4, (2007).
