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© 2018

Methods for Partial Differential Equations

Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models

Textbook

Table of contents

  1. Front Matter
    Pages i-xix
  2. Part I

    1. Front Matter
      Pages 1-1
    2. Marcelo R. Ebert, Michael Reissig
      Pages 3-5
    3. Marcelo R. Ebert, Michael Reissig
      Pages 7-15
    4. Marcelo R. Ebert, Michael Reissig
      Pages 17-35
    5. Marcelo R. Ebert, Michael Reissig
      Pages 37-48
    6. Marcelo R. Ebert, Michael Reissig
      Pages 49-55
    7. Marcelo R. Ebert, Michael Reissig
      Pages 57-68
    8. Marcelo R. Ebert, Michael Reissig
      Pages 69-75
  3. Part II

    1. Front Matter
      Pages 77-77
    2. Marcelo R. Ebert, Michael Reissig
      Pages 147-170
  4. Part III

    1. Front Matter
      Pages 171-171
    2. Marcelo R. Ebert, Michael Reissig
      Pages 173-179
    3. Marcelo R. Ebert, Michael Reissig
      Pages 181-189
    4. Marcelo R. Ebert, Michael Reissig
      Pages 191-226
    5. Marcelo R. Ebert, Michael Reissig
      Pages 227-239
    6. Marcelo R. Ebert, Michael Reissig
      Pages 241-269

About this book

Introduction

This book provides an overview of different topics related to the theory of partial differential equations. Selected exercises are included at the end of each chapter to prepare readers for the “research project for beginners” proposed at the end of the book. It is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area.

The book is organized in five parts:

In Part 1 the authors review the basics and the mathematical prerequisites, presenting two of the most fundamental results in the theory of partial differential equations: the Cauchy-Kovalevskaja theorem and Holmgren's uniqueness theorem in its classical and abstract form. It also introduces the method of characteristics in detail and applies this method to the study of Burger's equation.

Part 2 focuses on qualitative properties of solutions to basic partial differential equations, explaining the usual properties of solutions to elliptic, parabolic and hyperbolic equations for the archetypes Laplace equation, heat equation and wave equation as well as the different features of each theory. It also discusses the notion of energy of solutions, a highly effective tool for the treatment of non-stationary or evolution models and shows how to define energies for different models.

Part 3 demonstrates how phase space analysis and interpolation techniques are used to prove decay estimates for solutions on and away from the conjugate line. It also examines how terms of lower order (mass or dissipation) or additional regularity of the data may influence expected results.

Part 4 addresses semilinear models with power type non-linearity of source and absorbing type in order to determine critical exponents: two well-known critical exponents, the Fujita exponent and the Strauss exponent come into play. Depending on concrete models these critical exponents divide the range of admissible powers in classes which make it possible to prove quite different qualitative properties of solutions, for example, the stability of the zero solution or blow-up behavior of local (in time) solutions.

The last part features selected research projects and general background material.

Keywords

well-posedness of mixed problems and Cauchy problems regularity results potential theory energy method global existence of small data solutions source non-linearity absorbing non-linearity blow-up behavior hyperbolic systems

Authors and affiliations

  1. 1.Department of Computing and MathematicsUniversity of São PauloRibeirão PretoBrazil
  2. 2.Institute of Applied AnalysisTU Bergakademie FreibergFreibergGermany

About the authors

Marcelo Rempel Ebert (1977) is an Associate Professor at the Department of Computing and Mathematics at the University of São Paulo (USP). He obtained his Ph.D. degree in 2004 from Federal University of São Carlos, Brazil.  His original contributions are mainly devoted to Evolution  partial differential equations, in particular, questions related to the asymptotic behaviour and global existence of solutions for the Cauchy problem to semilinear wave equations.

Michael Gerhard Reissig (1958) is Professor for Partial Differential Equations at the Institute of Applied Analysis of the Technical University Bergakademie Freiberg. He obtained the degree Dr.rer.nat. in 1987, Dr.sc. in 1991 and Dr.habil. in 1992. His main contributions are devoted to the abstract Cauchy-Kovalevskaja theory, to Hele-Shaw flows, to elliptic equations, hyperbolic equations and Schrödinger equations as well.

Bibliographic information

  • Book Title Methods for Partial Differential Equations
  • Book Subtitle Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models
  • Authors Marcelo R. Ebert
    Michael Reissig
  • DOI https://doi.org/10.1007/978-3-319-66456-9
  • Copyright Information Springer International Publishing AG 2018
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-66455-2
  • Softcover ISBN 978-3-030-09772-1
  • eBook ISBN 978-3-319-66456-9
  • Edition Number 1
  • Number of Pages XVI, 456
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Partial Differential Equations
  • Buy this book on publisher's site
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Reviews

“This book contains both a careful presentation of several important theoretic notions and properties but also a selection of well-chosen exercises at the end of each chapter. … The exposition is flexible enough to allow substantial changes in the presentation of the arguments without compromising comprehension … . this volume is a valuable resource for advanced undergraduate and graduate students … . This book may also be useful for Ph.D. students or for special courses or seminars.” (Vicenţiu D. Rădulescu, Mathematical Reviews, October, 2018)​