Descript 
1 online resource (xvi, 456 pages) : illustrations, digital ; 24 cm 

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computer c rdamedia 

online resource cr rdacarrier 

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Note 
Part 1  Introduction  Part 2  Partial differential equations in models  Basics for partial differential equations  The CauchyKovalevskaja theorem  Holmgren's uniqueness theorem  Method of characteristics  Burger's equation  Laplace equation  properties of solutions  starting point of elliptic theory  Heat equation  properties of solutions  starting point of parabolic theory  Wave equation  properties of solutions  starting point of hyperbolic theory  Energies of solutions  one of the most important quantities  Part 3  Phase space analysis for heat equation  Phase space analysis and smoothing for Schrodinger equations  Phase space analysis for wave models  Phase space analysis for plate models  The method of stationary phase and applications  Part 4  Semilinear heat models  Semilinear classical damped wave models  Semilinear wave models with a special structural dissipation  Semilinear classical wave models  Semilinear Schrodinger models  Linear hyperbolic systems  Part 5  Research projects for beginners  Background material 

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 CauchyKovalevskaja 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 nonstationary 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 nonlinearity of source and absorbing type in order to determine critical exponents: two wellknown 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 blowup behavior of local (in time) solutions. The last part features selected research projects and general background material 
Host Item 
Springer eBooks

Subject 
Differential equations, Partial


Mathematics


Partial Differential Equations

Alt Author 
Reissig, Michael, author


SpringerLink (Online service)

