Overview
Provides an overview on different topics of the theory of partial differential equations
Presents a comprehensive treatment of semilinear models by using appropriate qualitative properties and a-priori estimates of solutions to the corresponding linear models and several methods to treat non-linearities
Supports the preparation of courses on selected topics of the theory of partial differential equations for advanced undergraduate and graduate students
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Table of contents (24 chapters)
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Front Matter
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Part II
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Front Matter
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Keywords
About this book
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.
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)
Authors and Affiliations
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
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2018
Hardcover ISBN: 978-3-319-66455-2Published: 06 March 2018
Softcover ISBN: 978-3-030-09772-1Published: 04 January 2019
eBook ISBN: 978-3-319-66456-9Published: 23 February 2018
Edition Number: 1
Number of Pages: XVI, 456
Number of Illustrations: 1 b/w illustrations
Topics: Partial Differential Equations