Winter semester 2016/17

Seminar: Blow-up 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 blow-up in ordinary differential and partial differential equations. Blow-up means, that solutions become unbounded in finite time, which leads to many interesting questions, e.g.

• 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 blow-up?

Topics

Blow-up of quadratic systems

• Existence of blow-up
• Asymptotic behaviour
• References: [1]

Blow-up criteria and blow-up rate estimates

• Concavity method to prove existence of blow-up
• Upper bound on blow-up rate
• References: [2], [3]

Blow-up in rescaled variables

• Self-similar variables
• Stationary solutions of rescaled flow
• Convergence to stationary solutions
• References: [2], [3], [4]

Blow-up from dynamical systems point of view

• Problem formulation
• Linearization and higher order asymptotics
• References: [5], [6], [7]

Description of the blow-up profile

• Description of blow-up profile for a single point
• References: [3]

Zero numbers and blow-up profile

• Zero numbers
• Construction of blow-up profile with n - blow-up points
• References: [2], [8]

Complete blow-up

• Definition of complete blow-up
• Existence of complete blow-up
• Existence of non-complete blow-up
• References: [2], [3]

References

• [1] Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormamlization group transformation. Discrete Contin. Dyn. Syst., Ser. B 21, pp. 437-446, (2016).
• [2] Bei Hu. Blow-up Theories for Semilinear Parabolic Equations. Springer, (2011).
• [3] Pavol Quittner, Phillippe Souplet. Superlinear Parabolic Problems Birkhäuser, (2007).
• [4] Yoshikazu Giga. Robert V. Kohn. Asymptotically Self-similar Blow-up of Semilinear Heat Equations. Communications on Pure and Applied Mathematics, Vol. XXXVIII, pp. 297-319, (1985).
• [5] Marek Fila, Hiroshi Matano. Blow-up in Nonlinear Heat Equations from the Dynamical Systems Point of View. Hanbook of Dynamical Systems, VOL. 2, pp. 723-758, (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 blow-up 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 Blow-up Profiles in One-dimensional Reaction Diffusion Equations. Journal of Dynamics and Differential Equations, Vol. 19, No. 4, (2007).