Abstract
We give an example of second-order hyperbolic operator for which the Cauchy problem is not Gevrey 6 well-posed for any lower order term. This phenomenon is caused by the existence of a null bicharacteristic landing on the double characteristic manifold. We also give an example of second-order hyperbolic operator for which the Cauchy problem is Gevrey 4 well-posed for any lower order term but not Gevrey s well-posed for s>4 with some lower order term.
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