Counterexamples to $C^{\infty}$ well posedness for some hyperbolic operators with triple characteristics

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Enrico Bernardi¹  Tatsuo Nishitani² 

[1] Dipartimento di Scienze Statistiche;Department of Mathematics, Osaka University

关键词: Cauchy problem;    well-posedness;    Gevrey class;    triple characteristics.;   

DOI  :  10.3792/pjaa.91.19

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

In this paper we prove a well posed and an ill posed result in the Gevrey category for a simple model hyperbolic operator with triple characteristics, when the principal symbol cannot be smoothly factorized, and whose propagation cone is not transversal to the triple characteristic manifold, thus confirming the conjecture that the Ivrii-Petkov condition is not sufficient for the $C^{\infty}$ well posedness unless the propagation cone is transversal to the characteristic manifold, albeit for a limited class of operators. Moreover we are able not only to disprove $C^{\infty}$ well posedness, but we can actually estimate the precise Gevrey threshold where well posedness will cease to hold.

【 授权许可】

Unknown