A Study of heteroclinic orbits for a class of fourth order ordinary differential equations


First Edition

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.


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Specifications


Publisher
Presses universitaires de Louvain
Title Part
Numéro 1
Author
Denis Bonheure,
Collection
Thèses de la Faculté des sciences
Language
English
Publisher Category
Natural Sciences > Mathematics > Mathematical Analysis
Publisher Category
Natural Sciences > Mathematics
Publisher Category
Natural Sciences
BISAC Subject Heading
SCI000000 SCIENCE
Onix Audience Codes
06 Professional and scholarly
CLIL (Version 2013-2019)
3051 SCIENCES FONDAMENTALES
Title First Published
01 January 2004
Type of Work
Thesis

Paperback


Publication Date
01 January 1947
ISBN-13
9782874634857
Extent
Main content page count : 78
Code
84709
Dimensions
16 x 24 x 0.5 cm
Weight
143 grams
List Price
12.00 €
ONIX XML
Version 2.1, Version 3

PDF


Publication Date
15 October 2011
ISBN-13
9782874637490
Extent
Main content page count : 80
Code
84709PDF
ONIX XML
Version 2.1, Version 3

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Contents


Préface vii

Avant-propos xi

Liste des publications lix

A Study of Heteroclinic Orbits for a Class of Fourth Order Ordinary Differential Equations 1

Contents 3

Introduction 5

An overview 7