Second order weakly hyperbolic operators with coefficients sum of powers of functions

We consider the Cauchy problem for the operator D-t(2) - D(x)a(t, x)D-x in the Gevrey classes. We show that if the coefficient a(t, x) is given by a finite sum of non negative functions then the Cauchy problem is well posed in the wider Gevrey class for the larger powers. We also give an example showing that the order of the Gevrey class obtained here is optimal.