Partial differential equations: time-periodic solutions

As far as the number of new results and quoted papers is concerned the present book may be considered a monograph. However, it also has some features of a textbook. Firstly, it proceeds from concrete problems to abstract ones, and secondly, all considerations and procedures are presented in much detail when met for the first time (such very elementary expositions can be found especially at the beginning of Chapters III and V). Finally, the authors focus their attention on elementary problems which can be dealt with by relatively simple methods. The authors hope that all this will make it possible also for an applied or technical research worker with some mathematical training to read this book. Naturally, the reader is supposed to be familiar with some basic notions from mathematical analysis, functional analysis and theory of partial differential equations. Also, the arguments and procedures which are repeated in the book are presented more briefly when met again, the reader being expected to become gradually more thoroughly acquainted with them. The authors have tried to provide a complete bibliography of all relevant publications (their number reaches about 500) from the theory of time-periodic solutions to non-linear partial and abstract differential equations whose origin may be put in the early thirties of this century.

I Preliminaries from Functional Analysis.- § 1. Linear Spaces and Operators.- §2. Function Spaces.- § 3. Existence Theorems for Operator Equations.- II Preliminaries from the Theory of Differential Equations.- § 1. Boundary-Value and Eigenvalue Problems for Elliptic and Ordinary Differential Operators.- § 2. The Wave and the Telegraph Equations.- § 3. The Heat Equation.- III The Heat Equation.- § 1. The (t, s)-Fourier Method.- § 2. The t-Fourier and s-Fourier Methods.- § 3. The Poincaré Method.- § 4. Supplements and Comments on the Linear and Weakly Non-Linear Heat Equation.- § 5. Comments on Strongly Non-Linear Parabolic Equations.- § 6. Comments on the Navier-Stokes Equations and Related Problems.- IV The Telegraph Equation.- § 1. The Fourier Methods.- § 2. The Poincaré Method.- § 3. Singularly Perturbed Problems.- § 4. Supplements and Comments on Linear and Weakly Non-Linear Problems.- § 5. Comments on Strongly Non-Linear Problems.- V The Wave Equation.- § 1. The Dirichlet Boundary Conditions; the Poincaré Method.- § 2. The Dirichlet boundary conditions; the Günzlcr method.- §3. Examples.- § 4. The Newton and Combined Boundary Conditions; the Günzler Method.- § 5. Entrainment of Frequency.- § 6. The Fourier Method.- § 7. The Wave Equation in an Unbounded Domain.- § 8. Supplements and Comments on Non-Autonomous Hyperbolic Equations.- § 9. Comments on Autonomous Hyperbolic Equations.- VI The Beam Equation and Related Problems.- § 1. The Equations of a Beam and of a Thin Plate.- § 2. Supplements and Comments.- § 3. The Dynamic von Kármán Equations of Thin Plates Involving Rotational Inertia and Damping.- VII The Abstract Equations.- § 1. The t-Fourier method: Preliminaries.- § 2. The t-Fourier Method: Spectral Properties of PeriodicOperators.- § 3. The t-Fourier Method: Weakly Non-Linear Problems.- § 4. Comments on Papers Using Direct Methods.- § 5. Comments on Papers using Indirect Methods.- Bibliography to Chapter I.- Bibliography to Chapter II.- Bibliography to Chapter III.- Bibliography to Chapter IV.- Bibliography to Chapter V.- Bibliography to Chapter VI.- Bibliography to Chapter VII.- Bibliography of papers on related topics.- Addenda to bibliography.- List of Symbols.