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Hyperbolic partial differential equations and wave phenomena. Transl. from the Japanese by Bohdan I. Kurpita. (English) Zbl 0948.35004

Translations of Mathematical Monographs. 189. Providence, RI: American Mathematical Society (AMS). xxi, 190 p. (2000).
This book is a translation from the Japanese original, published by Iwanami Shoten, Publishers, Tokyo, 1997.
It mainly concerns initial boundary value problems for second order linear hyperbolic equations. In Chapter one, such problems are developed, that describe transverse waves of strings, longitudinal waves of strings and propagation of sound, electromagnetic waves and elastic waves. Chapter two concerns qualitative solution properties, such as finite propagation of speed, domains of dependence, energy conservation and, mainly, existence and uniqueness (proved by means of the Hille-Yosida theorem) and regularity of the solutions. Chapter three deals with high frequency approximations of geometric optics, the Eikonal equation for the phase function, the transport equation for the amplitude function, reflection at the boundary and refraction at a surface of two different substances. Finally, Chapter four describes the decay in space and time of solutions in the exterior of a bounded space domain.
This small book is very carefully written, well organized and, hence, highly recommended for graduate students and researchers.
Reviewer: L.Recke (Berlin)

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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