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 Hormander [7] were sharp as enough as a sufficient condition, better than the Petrowsky condition, was conjectured. We here summarize them and sufficient conditions by V. Ya Ivrii, L. Hormander and the author. One of the conclusions for single equations is that the strong hyperbolicity is equivalent to the effective hyperbolicity.Remark. Throughout this paper, symbols of pseudodifferential operators are the Weyl symbols. An operator q(x, D) with the symbol q(x, £) is defined by q(x, D)<t> = (2n)-