Analysis of micro local structure of hyperbolic equations and characterization of hyperbolic equations for which the Cauchy problem is well-posed

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I considered the Cauchy problem for second-order hyperbolic operators with the coefficients of their principal parts depending only on the time variable. In the case where the coefficients are real analytic functions of the time variable, I obtained a sufficient condition for C^∞ well-posedness. And I showed that this sufficient condition is also a necessary one when the space dimension is less than 3 or the coefficients of the principal part are semi-algebraic functions (e. g., polynomials) of the time variable. Moreover, under this sufficient condition I proved that the singularities of solutions to the Cauchy problem propagate along broken null bicharacteristics.