Studies on a new class of hyperbolic systems

得到了双曲重特征线分类的一个确定性结果：如果双曲重特征线在参考点处的汉密尔顿映射只存在纯虚特征值，则称其为非有效双曲特征线.剩下的一个基本问题是围绕非有效双曲特征线的柯西问题是否是C-无限适定的？我们对双曲重特征线进行了分类，判定了双曲重特征线在参考重特征线周围的零特征线的行为是否相对于双特征线流形稳定，即双特征线流形中是否存在一个具有极限点的零特征线.我们得到了如下结果：如果零双特征线在参考双特征线周围的行为，则主符号是初等可分解的，柯西问题是C-无限适定的。另一方面，如果零双特征线的行为是不稳定的，那么主符号不是初等可分解的，柯西问题不是C-无限适定的。我们得到了更详细的结果。在这种不稳定情形下，Cauchy问题是Gevrey 5适定的，并且指数5在以下意义下是最优的：如果在双特征流形中存在一个具有极限点的零双特征线，则Cauchy问题对任何s> 5都不是Gevrey适定的。假设双特征流形的余维数为3，且汉密尔顿映射的所有特征值保持纯虚，则Cauchy问题是Gevrey 5适定的.

We have obtained a definitive result about the classification of hyperbolic double characteristics.A hyperbolic double characteristic is called non effectively hyperbolic characteristic if the Hamilton map at the reference point admits only pure imaginary eigenvalues. A remaining fundamental question was whether the Cauchy problem around non effectively hyperbolic characteristic is C-infty well-posed?We classify hyperbolic double characteristics whether the behavior of null bicharacteristics around the reference double characteristic is stable with respect to the doubly characteristic manifold, that is whether there exists a null bicharacteristic with a limit point in the doubly characteristic manifold. We have obtained the following results:If the behavior of null bicharacteristics around the reference double characteristic then the principal symbol is elementary decomposable and the Cauchy problem is C-infty well-posed. On the other hand, if the behavior of null bicharacteristic is unstable then the principal symbol is not elementary decomposable and the Cauchy problem is not C-infty well-posed. We obtained more detailed results. In this unstable case the Cauchy problem is Gevrey 5 well-posed and this index 5 is optimal in the following sense; if there is a null bicharacteristic with a limit point in the doubly characteristic manifold then the Cauchy problem is not Gevrey s well-posed for any s>5.Based on the above results, we obtained the following result : assume that the codimension of the doubly characteristic manifold is 3 and the all eigenvalues of the Hamilton map remain to be pure imaginary then the Cauchy problem is Gevrey 5 well-posed.