On the Cauchy problem for non-effectively hyperbolic operators: the Gevrey 3 well-posedness (Q2891093)

This paper deals with the Cauchy problem for second-order hyperbolic operator \(P\) with real analytic coefficients and principal symbol \(p\) vanishing exactly to order 2 on a smooth manifold \(\Sigma\). The Cauchy problem for \(P\) is \(C^\infty\) well-posed for any lower order term in the effectively hyperbolic case.NEWLINENEWLINEConsider now the non-effectively hyperbolic case, i.e. the Hamilton map \(F_p\) has no nonzero real eigenvalues on \(\Sigma\). As the case \(\text{Ker\,}F^2_p\cap \text{Im\,}F^2_p= 0\) on \(\Sigma\) is well studied, the authors of the paper assume that NEWLINE\[NEWLINE\text{Ker\,}F^2_p\cap \text{Im\,}F^2_p\neq 0\quad\text{on }\Sigma.\tag{\(*\)}NEWLINE\]NEWLINE Recently they proved that \((*)\), \(\text{codim\,}\Sigma=3\) and the assumption that there is no null integral curve of the Hamilton vector field \(H_p\) falling on \(\Sigma\) tangentially imply that the Cauchy problem for \(P\) is well-posed in \(G_s\), \(1\leq s< 4\) for any lower order term.NEWLINENEWLINEThis is the main result in that paper. Let \(\text{codim\,}\Sigma= 3\) and \((*)\) hold. Then the Cauchy problem for \(P\) is well-posed in \(G_s\), \(1\leq s<3\) for any lower order term. The Gevrey index 3 is optimal in appropriate sense.NEWLINENEWLINE A model example is given which verifies \((*)\), admits a null integral curve of \(H_p\) falling on \(\Sigma\) tangentially, but the Cauchy problem is not locally solvable for \(s> 3\).