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We investigate the Cauchy problem for second-order hyperbolic operators 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.

On the Cauchy Problem for Second-Order Hyperbolic Operators with the Coefficients of Their Principal Parts Depending Only on the Time Variable