Ill-Posed Cauchy Problem, Revisited
In Chap. 6 we exhibited a second order differential operator with polynomial coefficients for which the Cauchy problem is C ∞ ill-posed even though the Levi condition is satisfied. The Levi condition would be the most strict condition that one can impose on lower order terms on the double characteristics as far as we know. In this chapter we confirm this by proving that the Cauchy problem for this operator is ill-posed in the Gevrey class of order grater than 6 for any lower order term. In particular the Cauchy problem is C ∞ ill-posed for any lower order term. This phenomenon never occurs in the case of one spatial dimension. In the last section we give an example of second order differential operator of spectral type 1 on Σ, which shows that the IPH condition is not sufficient in general for the Cauchy problem to be C ∞ well-posed.
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