The Hyperbolic Cauchy Problem

1. Front Matter

   Pages I-VII

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2. Fourier integral operators with complex-valued phase function and the Cauchy problem for hyperbolic operators

   • Kunihiko Kajitani

   Pages 1-70

3. The effectively hyperbolic Cauchy problem

   • Tatsuo Nishitani

   Pages 71-167

4. Back Matter

   Pages 168-168

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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.

• Cauchy problem
• Derivative
• Hyperbolic operators
• differential equation
• hyperbolic partial differential equation
• partial differential equation
• partial differential equations
• phase function
• pseudo-differential operators