Necessary conditions for hyperbolic systems
This paper deals with the Cauchy problem for first order hyperbolic system \(L(x,{\mathcal D})\) having characteristic points of higher multiplicity. In other words, the determinant of the principal symbol has multiple characteristic points. The authors consider the case when at a multiple characteristic point the principal symbol of the system has corank 2. The main aim here is to obtain general necessary conditions for the \(C^\infty\) well posedness of the Cauchy problem. These conditions involve a suitably defined determinant \(\text{Det}_{(s)}\) of the full symbol of \(L\). The method of the proof consists in constructing of an asymptotic null solution which violates an a priori estimate resulting from the \(C^\infty\) well posedness of the Cauchy problem.
scientific article; zbMATH DE number 1886860