Counterexamples to $ 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. 
Enrico Bernardi, Tatsuo Nishitani
math.AP