The Cauchy Problem for Hyperbolic Equations with Double Characteristics

Abstract

In the last decade, the theory on the Cauchy problem for hyperbolic equations has developed a little in the analysis of their characters. This article will aim to survey briefly the main point of the advance. Before doing so, we give a short historical remark. The notion of hyperbolic equation began from the characterization of the wave equation. In the present day, however, it comes to be understood as an algebraic and geometric characterization, for symbols of partial differential operators, corresponding to solvability of non characteristic Cauchy problem for them with data of a suitable function space, so called Especially it seems to have been considered that the solvability to the space of infinitely difYerentiable functions called <f-well posed is essential since J. Hadamar [5] proposed. In fact I. G. Petrowsky [24] and completely L. Carding [8] characterized the necessary and sufficient condition for it to be cfwell posed in case of constant coefficients. In general, namely, to the equations with variable coefficients, P. D. Lax [14] and S. Mizohata [19] got a necessary condition that the principal part of the equations should be hyperbolic if the Cauchy problem for it is well posed. Nevertheless the generic fact known about the sufficient condition invariant under the change of variables had been essentially only one given by I. G. Petrowsky [25] for a long time. The fact is that the Cauchy problem for a hyperbolic equation with simple characteristics, called strictly hyperbolic, is <f-well posed, although it was extended until uniformly symmetrizable ones in case of systems of operators. The case of constant coefficients and the results by A. Lax [13] and by M. Yamaguti [28] have indicated the necessity of clarifying relations between the principal part and the lower