# Cauchy Problem in the Gevrey Classes

Chapter

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## Abstract

In Chap. 6 we showed that there exists a second order differential operator of spectral type 2 on *Σ* with bicharacteristics tangent to the double characteristic manifold for which the Cauchy problem is ill-posed in the Gevrey class of order *s* for any *s* > 5 even though the Levi condition is satisfied. The best we can expect is the well-posedness in the Gevrey class of order 5 under the Levi condition. This is indeed the case. We prove that for general second order differential operator of spectral type 2 on *Σ* which may have tangent bicharacteristics, the Cauchy problem is well-posed in the Gevrey class of order 5 under the Levi condition.

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