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Keywords:

  • wave equation;
  • initial–boundary-value problems;
  • scattering theory;
  • historical remark

Abstract

A lot of vibration processes in mathematical physics are described by the wave equation or by related equations and systems, and plenty of research has been done on this subject. The results and methods obtained thereby have been very important in other fields of application, and they still are. They also had and still have an immense influence on the development of mathematics as a whole.

The following survey tries to convey an impression of how exciting the research in this field of partial differential equations has been in the 20th century, and it wants to present some of the interesting results achieved. The selection, of course, reflects personal tastes and interests. It starts reporting on the state of the art at the end of the last century. It then describes important solution methods typical for this century, as there are integral equation methods, Hilbert space methods, or spectral representation. Generalized solutions to initial–boundary-value problems are mentioned, and some applications are indicated, e.g. on the asymptotic behaviour of solutions, in scattering theory, and in non-linear analysis. Copyright © 2001 John Wiley & Sons, Ltd.