On the well posedness of the Cauchy problem for a class of hyperbolic operators with multiple involutive characteristics (Q1356384)

Let \(X\subset\mathbb{R}^{n+1}= \mathbb{R}_{x_0}\times \mathbb{R}^n_{x_1}\) be an open set such that \(0\in X\), and let \(P(x,D)= P_m(x,D)+ P_{m-1}(x,D)+\cdots\) be a differential operator of order \(m\) with \(C^\infty\) coefficients. The hyperplane \(x_0=0\) is non-characteristic for \(P\), the principal symbol \(P_m\) is hyperbolic with respect to \(x_0\), \(P_m\) vanishes exactly of order \(m_1\leq m\) on a closed, conic, non-radial involutive submanifold \(\Sigma\) of codimension \(d+1\) in \(T^*X\backslash 0\), being strictly hyperbolic outside \(\Sigma\), and the localization of \(P_m\) at \(\rho\) (for any \(\rho\in\Sigma\)) is hyperbolic with respect to a certain direction. The result is the following: Under these assumptions, the Cauchy problem for \(P\) is well-posed in \(X\) iff a certain Levi condition holds.