Hyperbolic equations with non-analytic coefficients

We prove that the Cauchy problem for a hyperbolic, homogeneous equation with C-infinity coefficients depending on time, is well posed in every Gevrey class, although in general it is not well-posed in C-infinity, provided the characteristic roots satisfy the condition lambda(i)(t)(2) +lambda(j)(t)(2) <= M(lambda(i)(t) -lambda(j)(t))(2) (i not equal j).