Notes on tangent bicharacteristics and ill-posedness of the Cauchy problem

TL;DR

This paper investigates a class of second-order hyperbolic operators with spectral transitions, demonstrating that the Cauchy problem can be ill-posed in certain Gevrey classes due to tangent bicharacteristics.

Contribution

It introduces specific hyperbolic operators with spectral transitions and proves their associated Cauchy problems are ill-posed in particular Gevrey classes, highlighting tangent bicharacteristics' role.

Findings

Cauchy problem is not locally solvable in certain Gevrey classes.

Operators exhibit spectral transition of the Hamilton map.

Presence of tangent bicharacteristics influences ill-posedness.

Abstract

We exhibit a family of second-order hyperbolic differential operators presenting spectral transition of the Hamilton map. As a consequence we prove that the Cauchy problem is not locally solvable at the origin in Gevrey classes of order greater than some fixed value. The main feature of these operators is that they may all have bicharacteristics tangent to the double manifold.