The Hyperbolic Cauchy Problem

The approach to the Cauchy problem taken here by the authors

is based on theuse of Fourier integral operators with a

complex-valued phase function, which is a time function

chosen suitably according to the geometry of the multiple

characteristics. The correctness of the Cauchy problem in

the Gevrey classes for operators with hyperbolic principal

part is shown in the first part. In the second part, the

correctness of the Cauchy problem for effectively hyperbolic

operators is proved with a precise estimate of the loss of

derivatives. This method can be applied to other (non)

hyperbolic problems. The text is based on a course of

lectures given for graduate students but will be of interest

to researchers interested in hyperbolic partial differential

equations. In the latter part the reader is expected to be

familiar with some theory of pseudo-differential operators.