Methods for Partial Differential Equations: Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models

Description

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. Front Matter ....Pages i-xix Front Matter ....Pages 1-1 Introduction (Marcelo R. Ebert, Michael Reissig)....Pages 3-5 Partial Differential Equations in Models (Marcelo R. Ebert, Michael Reissig)....Pages 7-15 Basics for Partial Differential Equations (Marcelo R. Ebert, Michael Reissig)....Pages 17-35 The Cauchy-Kovalevskaja Theorem (Marcelo R. Ebert, Michael Reissig)....Pages 37-48 Holmgren’s Uniqueness Theorem (Marcelo R. Ebert, Michael Reissig)....Pages 49-55 Method of Characteristics (Marcelo R. Ebert, Michael Reissig)....Pages 57-68 Burgers’ Equation (Marcelo R. Ebert, Michael Reissig)....Pages 69-75 Front Matter ....Pages 77-77 Laplace Equation—Properties of Solutions—Starting Point of Elliptic Theory (Marcelo R. Ebert, Michael Reissig)....Pages 79-101 Heat Equation—Properties of Solutions—Starting Point of Parabolic Theory (Marcelo R. Ebert, Michael Reissig)....Pages 103-117 Wave Equation—Properties of Solutions—Starting Point of Hyperbolic Theory (Marcelo R. Ebert, Michael Reissig)....Pages 119-145 The Notion of Energy of Solutions: One of the Most Important Quantities (Marcelo R. Ebert, Michael Reissig)....Pages 147-170 Front Matter ....Pages 171-171 Phase Space Analysis for the Heat Equation (Marcelo R. Ebert, Michael Reissig)....Pages 173-179 Phase Space Analysis and Smoothing for Schrödinger Equations (Marcelo R. Ebert, Michael Reissig)....Pages 181-189 Phase Space Analysis for Wave Models (Marcelo R. Ebert, Michael Reissig)....Pages 191-226 Phase Space Analysis for Plate Models (Marcelo R. Ebert, Michael Reissig)....Pages 227-239 The Method of Stationary Phase and Applications (Marcelo R. Ebert, Michael Reissig)....Pages 241-269 Front Matter ....Pages 271-271 Semilinear Heat Models (Marcelo R. Ebert, Michael Reissig)....Pages 273-297 Semilinear Classical Damped Wave Models (Marcelo R. Ebert, Michael Reissig)....Pages 299-324 Semilinear Wave Models with a Special Structural Dissipation (Marcelo R. Ebert, Michael Reissig)....Pages 325-349 Semilinear Classical Wave Models (Marcelo R. Ebert, Michael Reissig)....Pages 351-365 Semilinear Schrödinger Models (Marcelo R. Ebert, Michael Reissig)....Pages 367-382 Linear Hyperbolic Systems (Marcelo R. Ebert, Michael Reissig)....Pages 383-401 Front Matter ....Pages 403-403 Research Projects for Beginners (Marcelo R. Ebert, Michael Reissig)....Pages 405-421 Background Material (Marcelo R. Ebert, Michael Reissig)....Pages 423-463 Back Matter ....Pages 465-479