DFG Collaborative Research Center 647 on
Space, time, matter

Research Project

Singularity structure, longtime behaviour and dynamics of solutions of nonlinear evolution equations

Research directors

Prof. Dr. Klaus Ecker

Prof. Dr. Bernold Fiedler

Members

Dr. Stefan Liebscher

Dr. Juliette Hell

Tobias Marxen

Summary

The project aims at a systematic understanding of the interplay between geometry and dynamics of specific partial differential equations. Emphasis is on the global structure of ancient, immortal, and eternal solutions which exist globally for backward, forward, and all time, respectively. Of particular interest are geometric and topological descriptions of the mechanisms which lead to blow-up in finite time, and to a meaningful continuation beyond blow-up.

Zusammenfassung

Das Projekt zielt auf ein systematisches Verständnis der Wechselwirkung zwischen der Geometrie und der Dynamik konkreter partieller Differentialgleichungen. Im Vordergrund steht die globale Struktur antiker, unsterblicher und ewiger Lösungen, die global existieren für negative, positive bzw. alle Zeiten. Besonderes Interesse gilt geometrischen und topologischen Beschreibungen der Mechanismen, die zu blow-up in endlicher Zeit führen und eine sinnvolle Fortsetzung über diesen Zeitpunkt hinaus erlauben.

Preprints

Marc Georgi:

The shape of profiles exhibiting blow up in forward and backward time
Preprint, Freie Universität Berlin, 2007.
(Preprint, PDF)    

B. Fiedler, C. Rocha

Connectivity and Design of Planar Global Attractors of Sturm Type. I: Bipolar Orientations and Hamiltonian Paths
J. Reine Angew. Math., 2008, in press.
(pdf, 750 kB)

B. Fiedler, C. Rocha

Connectivity and Design of Planar Global Attractors of Sturm Type. II: Connection Graphs.
J. Differential equations,2008.
(pdf, 4.6 MB)

B. Fiedler, C. Rocha

Connectivity and Design of Planar Global Attractors of Sturm Type. III: Small and Platonic Examples.
Submitted, 2007.
sections 1&2 (pdf, 2.4 MB), section 3 (pdf, 5.6 MB), section 4 (pdf, 5.7 MB).

B. Fiedler, H. Matano

Global dynamics of blow-up profiles in one-dimensional reaction diffusion equations
J.Dyn.Diff.Eqs., 2007.
(pdf, 250 kB)