  We discuss the well-posedness of the Cauchy problem for hyperbolic operators
with double characteristics which changes from non-effectively hyperbolic to
effectively hyperbolic, on the double characteristic manifold, across a
submanifold of codimension 1. We assume that there is no bicharacteristic
tangent to the double characteristic manifold and the spatial dimension is 2.
Then we prove the well-posedness of the Cauchy problem in all Gevrey classes
assuming, on the double characteristic manifold, that the ratio of the
imaginary part of the subprincipal symbol to the real eigenvalue of the
Hamilton map is bounded and that the sum of the real part of the subprincipal
symbol and the modulus of the imaginary eigenvalue of the Hamilton map is
strictly positive.
