Tangent Bicharacteristics and Ill-Posedness

In this chapter we provide a homogeneous second order differential operator P of spectral type 2 on Sigma ith polynomial coefficients with a bicharacteristic tangent to the double characteristic manifold and satisfies the Levi condition for which the Cauchy problem is ill-posed in the Gevrey class of order s > 5, in particular the Cauchy problem is C-infinity ill-posed. We also discuss some open questions on the Cauchy problem for P of spectral type 2 with tangent bicharacteristics and no transition. In the last section we make some remarks on the Cauchy problem for P with transition of spectral type under the assumption of no tangent bicharacteristics.