On the Cauchy problem for hyperbolic operators of second order whose coefficients depend only on the time variable

Abstract

In this paper we deal with hyperbolic operators of second order whose coefficients depend only on the time variable and give necessary conditions and sufficient conditions for the Cauchy problem to be $C^{\infty }$ well-posed. In particular, we give a necessary and sufficient condition (a complete characterization) for $C^{\infty }$ well-posedness when the space dimension is equal to 2 and the coefficients are real analytic functions of the time variable.