For second-order hyperbolic operators with a spectral transition of the Hamilton map and tangent bicharacteristics, the Cauchy problem is not locally solvable in Gevrey classes of order greater than max{5,1+N/l}.
Springer, Cham, 2017
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Notes on tangent bicharacteristics and ill-posedness of the Cauchy problem
For second-order hyperbolic operators with a spectral transition of the Hamilton map and tangent bicharacteristics, the Cauchy problem is not locally solvable in Gevrey classes of order greater than max{5,1+N/l}.