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Iwanami Series in Modern Mathematics
Hyperbolic Partial Differential Equations and Wave Phenomena
Mitsuru Ikawa, Kyoto University, Japan
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Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics
2000; 190 pp; softcover
Volume: 189
ISBN-10: 0-8218-1021-9
ISBN-13: 978-0-8218-1021-7
List Price: US$42
Member Price: US$33.60
Order Code: MMONO/189
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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

"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

  • Wave phenomena and hyperbolic equations
  • The existence of a solution for a hyperbolic equation and its properties
  • The construction of asymptotic solutions
  • Local energy of the wave equation
  • Perspectives on current research in mathematics
  • Bibliography
  • Solutions to the exercises
  • Index
  • Copy/reprinting (cont'd)
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