| Abstract: |
| Parabolic partial differential equations are fundamental models for many
applied phenomena. For example, pattern formation in diblock copolymers
is described by the evolution of the fourth-order Ohta-Kawasaki equation,
while competition in the interaction of populations can be modeled by
second-order reaction-diffusion systems such as the Shigesada-Kawasaki-Teramoto
model. Essential for a deeper understanding of the long-term dynamics of such
problems are the sets of equilibria as described by the associated bifurcation
diagrams. In this talk, we present computer-assisted proof techniques which can
be used to validate and continue bifurcation points through the use of suitable
extended systems. This includes not only fold points, but also pitchfork and
transcritical bifurcations which are the result of group actions beyond forcing
through involutions. Our results apply to both one- and two-dimensional domains,
and they can also be used to treat certain bifurcation points with
higher-dimensional kernels. |
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