Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Μπίρμπα Παναγιώτα
Supervisors info:
Καθηγητής Νικόλαος Αλικάκος (επιβλέπων)
Original Title:
Reaction-diffusion problems and finite dimensional dynamical system
Translated title:
Προβλήματα αντίδρασης-διάχυσης και δυναμικά συστήματα πεπερασμένων διαστάσεων
Summary:
At the current M.Sc thesis we study Reaction-Di usion Problems and Finite
Dimensional Dynamical Systems. Especially, we penetrate in several results that
are connected with the one dimensional scalar Allen-Cahn equation as well as
one dimensional vector Allen-Cahn equation. In chapter 1 we make an
introduction to the basic concepts of the O.D.E Theory that are useful,
presenting the heteroclinic solutions as well as their asymptotic behavior and
the instanton. Furthermore, we analyze basic spectral facts for specific
linearized operators. We proceed in chapter 2 studying the slow motion via
energy and spectrum in one dimensional space with a presentation based on [1].
We provide several notices and proofs that lead us to present a central result
in the developed object. In chapter 3 we study slow motion manifolds for a
class of singular perturbation. More speci cally, in section 3.1 we based on
the work [3] in order to present two crucial results, noticing all the
techniques that are developed and their connection with the physical
phenomenon. Additionally, in section 3.2 we examine analytically one of the two
basic stages of the slow motion, which is usually referred as separation.
Finally we conclude with Chapter 4 where we display our attempt to generalize
the work of J. Neu [18] which can be characterized by a method of formal
asymptotics in one dimensional scalar case.
Keywords:
Heteroclinic Solution, Slow motion Manifolds, spectral gap, Formal Asymptotics, Allen-Cahn equation
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