On the Cauchy Problem for Hyperbolic Operators with Double Characteristics, a Transition Case
We discuss the well-posedness of the Cauchy problem for hyperbolic operators with double characteristics when the operator is effectively hyperbolic outside a submanifold of codimension 1 of the double characteristic manifold. We assume that there is no null bicharacteristic tangent to the double characteristic manifold. We derive microlocal weighted energy estimates which proves the well-posedness of the Cauchy problem in all Gevrey classes assuming that the ratio of the imaginary part of the subprincipal symbol to the real eigenvalue of the Hamilton map is bounded on the double characteristic manifold and that the strict Ivrii-Petkov-Hörmander condition holds on the transition manifold.
KeywordsCauchy problem well-posedness effectively hyperbolic transitioncase null bicharacteristics
Unable to display preview. Download preview PDF.