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Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II

Applications

  • Book
  • © 2021

Overview

Authors:
  1. Atsushi Yagi

    1. Osaka University, Suita, Osaka, Japan

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  • Demonstrates the asymptotic convergence to stationary solutions for global solutions of abstract parabolic equations
  • Includes n-dimensional semilinear parabolic equations and higher dimensional Keller–Segel equations among its topics
  • Provides the methodology for presenting extremely precise convergence results

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-ix
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  2. Preliminaries

    • Atsushi Yagi
    Pages 1-27

  3. Review of Abstract Results

    • Atsushi Yagi
    Pages 29-38

  4. Parabolic Equations

    • Atsushi Yagi
    Pages 39-69

  5. Epitaxial Growth Model

    • Atsushi Yagi
    Pages 71-90

  6. Chemotaxis Model

    • Atsushi Yagi
    Pages 91-120

  7. Back Matter

    Pages 121-128
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Keywords

About this book

This second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz–Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described.

Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller–Segel equations in one-, two-, and three-dimensionalspaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more.


Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.

Authors and Affiliations

  • Osaka University, Suita, Osaka, Japan

    Atsushi Yagi

Bibliographic Information

  • Book Title: Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II

  • Book Subtitle: Applications

  • Authors: Atsushi Yagi

  • Series Title: SpringerBriefs in Mathematics

  • DOI: https://doi.org/10.1007/978-981-16-2663-0

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021

  • Softcover ISBN: 978-981-16-2662-3Published: 13 August 2021

  • eBook ISBN: 978-981-16-2663-0Published: 12 August 2021

  • Series ISSN: 2191-8198

  • Series E-ISSN: 2191-8201

  • Edition Number: 1

  • Number of Pages: IX, 128

  • Number of Illustrations: 607 b/w illustrations

  • Topics: Analysis, Functional Analysis, Measure and Integration

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