Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed

In this article we study the $C^\infty$-well-posedness of the non-characteristic Cauchy problem for hyperbolic equations with characteristic roots of variable multiplicity. We obtain a necessary condition for the Cauchy problem with arbitrary lower order terms to be well-posed, and also a necessary condition for the smoothness of the solution to be independent of the lower order terms. For equations with characteristic roots of an arbitrary variable multiplicity we obtain necessary conditions on the lower order terms for the Cauchy problem to be well-posed. All the proofs are based on a single method: the construction of asymptotic solutions.