Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Supervisors info:
Ιωάννης Στρατής, Καθηγητής, τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Νικόλαος Αλικάκος, Καθηγητής, τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Νικόλαος Καραχάλιος, Καθηγητής, τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου
Original Title:
Semilinear Parabolic Evolution Equations and an Application to the Lotka-Volterra System with Diffusion
Translated title:
Semilinear Parabolic Evolution Equations and an Application to the Lotka-Volterra System with Diffusion
Summary:
The aim of this Master’s thesis is the study of some basic methods and theoretical results obtained in the last decades, concerning time dependent (evolution) semilinear equations (the nonlinearity appearing only in the lower order terms) and systems of parabolic type.
A useful tool for this kind of evolution problems will be linear one-parameter Operator Semigroups, whose basic theory will concern us in the first chapter of this dissertation. In particular, we find conditions under which a, generally unbounded, operator in a Banach space generates a semigroup and conversely, and also relate the generated semigroup with the solution of the corresponding abstract Cauchy problem.
In the second chapter we use the results of the previous chapter to prove theorems concerning the solutions of the semilinear Cauchy problem (local/global existence, uniqueness, regularity).
The third chapter is devoted to the heat equation, the typical representative of the class of parabolic equations. After a brief study of the linear case, we move on to the semilinear equation. In paragraph 3.4 we find conditions for global existence for "small enough" and "large enough" initial data. Of extreme importance here will be the form of the maximum principle given at the beginning of the paragraph. In paragraph 3.5 we present two typical methods for proving blow-up of solutions in finite time.
In the last chapter we study some basic properties of the solutions of a fairly general Reaction-Diffusion system, in the context of ecological models (predator-prey), and we proceed to specialize these results to the Lotka-Volterra system with diffusion. The results of this chapter are based on the work of N.Alikakos (1979) and improve some previous related results by Chow and Williams (1978). The basic theorem that enables us to do so is the a priori estimate at paragraph 4.3.
Some simple definitions and theorems from the theory of Dynamical Systems that we used in the last chapter are contained in the Appendix at the end of this thesis.
Main subject category:
Science
Other subject categories:
Analysis
Keywords:
Operator Semigroups, Differential Equations of parabolic type, Lotka-Volterra system
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Master's thesis.pdf
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