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 Oleinik [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 u(u), - Sigma(n)(i,j=1)(a(ij)(t, x)u(xj))(xi) = 0, with Sigma a(ij)xi(i)xi(j) >= 0,. 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 x 2 systems.