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Methods for Partial Differential Equations

Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models

Autor: Marcelo R. Ebert, Michael Reissig

Limbă:

Legare: Copertă tare

Editura: Birkhauser Verlag AG

Disponibilitate: În depozitul extern

Expediem în 10-13 zile

488.04 lei

This book provides an overview of different topics related to the theory of partial differential equ...

Informații despre carte

Autor

Marcelo R. Ebert, Michael Reissig

Limbă

engleză

Legare

Carte - Copertă tare

Publicat

2018

Pagini

456

EAN

9783319664552

ISBN

3319664557

Enbook ID

16803739

Editura

Birkhauser Verlag AG

Greutate

896

Dimensiuni

165 x 244 x 33

Descriere completă

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. 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 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 semi-linear 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. The book is a valuable resource for advanced graduates and undergraduate students who are interested in specializing in this area.