Abstract Parabolic Evolution Equations and their Applications

Abstract Parabolic Evolution Equations and their Applications


The semigroup methods are known as a powerful tool for analyzing nonlinear diffusion equations and systems. The author has studied abstract parabolic evolution equations and their applications to nonlinear diffusion equations and systems for more than 30 years. He gives first, after reviewing the theory of analytic semigroups, an overview of the theories of linear, semilinear and quasilinear abstract parabolic evolution equations as well as general strategies for constructing dynamical systems, attractors and stable-unstable manifolds associated with those nonlinear evolution equations.

In the second half of the book, he shows how to apply the abstract results to various models in the real world focusing on various self-organization models: semiconductor model, activator-inhibitor model, B-Z reaction model, forest kinematic model, chemotaxis model, termite mound building model, phase transition model, and Lotka-Volterra competition model. The process and techniques are explained concretely in order to analyze nonlinear diffusion models by using the methods of abstract evolution equations.

Thus the present book fills the gaps of related titles that either treat only very theoretical examples of equations or introduce many interesting models from Biology and Ecology, but do not base analytical arguments upon rigorous mathematical theories.

Table of contents

Front Matter....Pages I-XVIII
Preliminaries....Pages 1-54
Sectorial Operators....Pages 55-116
Linear Evolution Equations....Pages 117-176
Semilinear Evolution Equations....Pages 177-200
Quasilinear Evolution Equations....Pages 201-249
Dynamical Systems....Pages 251-315
Numerical Analysis....Pages 317-344
Semiconductor Models....Pages 345-356
ActivatorInhibitor Models....Pages 357-371
BelousovZhabotinskii Reaction Models....Pages 373-389
Forest Kinematic Model....Pages 391-415
Chemotaxis Models....Pages 417-443
Termite Mound Building Model....Pages 445-470
Adsorbate-Induced Phase Transition Model....Pages 471-486
LotkaVolterra Competition Model with Cross-Diffusion....Pages 487-526
Characterization of Domains of Fractional Powers....Pages 527-562
Back Matter....Pages 563-581


  • Author: Atsushi Yagi (auth.)
  • Edition: 1
  • Publication Date: 2010
  • Publisher: Springer-Verlag Berlin Heidelberg
  • ISBN-10: 3642046304
  • ISBN-13: 9783642046308
  • Pages: 594
  • Format: pdf
  • Size: 4.1M
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