A result on the well posedness of the Cauchy problem for a class of hyperbolic operators with double characteristics (Q1904090)

Summary: Let \(p_2\) be the principal symbol of a hyperbolic differential operator \(P\) of order two admitting characteristic roots of variable multiplicity. Suppose that the double characteristic manifold \(\Sigma\) of \(p_2\) contains a submanifold \(\widetilde {\Sigma}\) such that at each point of \(\widetilde {\Sigma}\) the Hamiltonian matrix of \(p_2\), \(F\), has a Jordan block of dimension 4, whereas at each point of \(\Sigma\setminus \widetilde {\Sigma}\), \(F\) admits only Jordan blocks of size 2 and \(F\) is not effectively hyperbolic. We prove that under suitable conditions on the 3-jet of \(p_2\) at \(\widetilde {\Sigma}\) the Cauchy problem for \(P\) is well posed provided the usual Levi conditions on the lower order terms are satisfied.