On the Cauchy Problem for Hyperbolic Operators with Double Characteristics whose Principal Parts Have Time Dependent Coefficients

Abstract

In this paper we investigate the Cauchy problem for hyperbolic operators with double characteristics in the framework of the space of C∞ functions. In the case where the coefficients of their principal parts depend only on the time variable and are real analytic, we give a sufficient condition for C∞ well-posedness, which is also a necessary one when the space dimension is less than 3 or the coefficients of the principal parts are semi-algebraic functions ( e.g., polynomials) of the time variable.