Advertisement

Methods for Partial Differential Equations

Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models

  • Marcelo R. Ebert
  • Michael Reissig

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
  5. Part IV

    1. Front Matter
      Pages 271-271
    2. Marcelo R. Ebert, Michael Reissig
      Pages 273-297
    3. Marcelo R. Ebert, Michael Reissig
      Pages 299-324
    4. Marcelo R. Ebert, Michael Reissig
      Pages 325-349
    5. Marcelo R. Ebert, Michael Reissig
      Pages 351-365
    6. Marcelo R. Ebert, Michael Reissig
      Pages 367-382
    7. Marcelo R. Ebert, Michael Reissig
      Pages 383-401
  6. Part V

    1. Front Matter
      Pages 403-403
    2. Marcelo R. Ebert, Michael Reissig
      Pages 405-421
    3. Marcelo R. Ebert, Michael Reissig
      Pages 423-463
  7. Back Matter
    Pages 465-479

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

  • Marcelo R. Ebert
    • 1
  • Michael Reissig
    • 2
  1. 1.Department of Computing and MathematicsUniversity of São PauloRibeirão PretoBrazil
  2. 2.Institute of Applied AnalysisTU Bergakademie FreibergFreibergGermany

Bibliographic information