**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2000; 190 pp; Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-1021-7

Product Code: MMONO/189

Print ISBN: 978-0-8218-1021-7

Product Code: MMONO/189

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

**Electronic ISBN: 978-1-4704-4603-1
Product Code: MMONO/189.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

# Hyperbolic Partial Differential Equations and Wave Phenomena

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*Mitsuru Ikawa*

The familiar wave equation is the most fundamental
hyperbolic partial differential equation. Other hyperbolic equations,
both linear and nonlinear, exhibit many wave-like phenomena. The
primary theme of this book is the mathematical investigation of such
wave phenomena.

The exposition begins with derivations of some wave equations,
including waves in an elastic body, such as those observed in
connection with earthquakes. Certain existence results are proved
early on, allowing the later analysis to concentrate on properties of
solutions. The existence of solutions is established using methods
from functional analysis. Many of the properties are developed using
methods of asymptotic solutions. The last chapter contains an
analysis of the decay of the local energy of solutions. This analysis
shows, in particular, that in a connected exterior domain,
disturbances gradually drift into the distance and the effect of a
disturbance in a bounded domain becomes small after sufficient time
passes.

The book is geared toward a wide audience interested in PDEs.
Prerequisite to the text are some real analysis and elementary
functional analysis. It would be suitable for use as a text in PDEs or
mathematical physics at the advanced undergraduate and graduate
level.

#### Readership

Advanced undergraduate and graduate students and researchers interested in partial differential equations and mathematical physics.

#### Reviews & Endorsements

Interesting and welcome little book … provides an excellent introduction to the mathematics of linear wave propagation … This book provides a good introduction to the fascinating topic of linear wave propagation for a reader who has a sound mathematical background but no special familiarity with either the physical or mathematical aspects of wave propagation phenomena.

-- Mathematical Reviews

This small book is very carefully written, well-organized, and hence, highly recommended for graduate students and researchers.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Hyperbolic Partial Differential Equations and Wave Phenomena

- Cover Cover11
- Title page v6
- Contents vii8
- Preface to the English edition xi12
- Preface to the Japanese edition xiii14
- Outline of the theory and objectives xvii18
- Wave phenomena and hyperbolic equations 124
- The existence of a solution for a hyperbolic equation and its properties 3154
- The construction of asymptotic solutions 83106
- Local energy of the wave equation 153176
- Perspectives on current research in mathematics 175198
- Bibliography 179202
- Solutions to the exercises 181204
- Index 189212
- Copy/reprinting (cont’d) 191214
- Back Cover Back Cover1216