Counterexamples to C∞ C^{\infty} well posedness for some hyperbolic operators with triple characteristics

In this paper we prove that for a class of non-effectively hyperbolic operators with smooth triple characteristics the Cauchy problem is well posed in the Gevrey 2 class, beyond the generic Gevrey class $3/2$ (see e.g. \cite{Bro}). Moreover we show that this value is optimal

On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness

2000 Mathematics Subject Classification: 35L15, Secondary 35L30.In this paper we prove that for non effectively hyperbolic operators with smooth double characteristics with the Hamilton map exhibiting a Jordan block of size 4 on the double characteristic manifold the Cauchy problem is well posed in the Gevrey 6 class if the strict Ivrii-Petkov-Hörmander condition is satisfied