Some results on the well-posedness for second order linear equations

We investigate the Cauchy problem for second order hyperbolic equations of complete form, and we prove an extension of a classical result of Ole?inik [10] concerning the well-posedness for equations in which are absent the terms with mixed time-space derivatives. Then, in space dimension n = 1, we compare our results with those in [8] for equations with analytic coefficients, and those of [7] and [11] for homogeneous equations with coefficients depending only either on t or on x. Moreover we exhibit, in space dimension n≥ 2, an equation of the form where the coefficients are analytic functions, for which the Cauchy problem is ill-posed. Finally, we present a sufficient condition for the well-posedness of 2× 2 systems.