The Cauchy Problem for Uniformly Diagonalizable Hyperbolic Systems in Gevrey Classes

Abstract

This chapter discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems of linear partial differential equations in Gevrey classes. It also discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems whose coefficients are in Gevrey class. It has been proven that if the coefficients of the system are constant, the uniformly diagonalizable hyperbolic system is equivalent to strongly hyperbolic one, that is, this system is stable under the perturbation of lower order term of the system. In general, the characterization of the strong hyperbolicity in the C∞-sense for the systems of variable coefficients is an open problem. The uniform diagonalizability for the hyperbolic systems is a sufficient condition for the Cauchy problem to be well-posed.