The Cauchy problem (variable coefficients)

Abstract

In this chapter we shall extend the existence theory of section 5.6 to operators with variable coefficients which are strictly hyperbolic at every point. It is possible to modify the proof of Theorem 5.4.1 by using some of the techniques in the proof of Theorem 6.1.1 in order to show that the Cauchy problem for the operator P cannot be solved for arbitrary data unless the principal part of P is hyperbolic in the initial surface (at least if there are no multiple characteristics). However, we shall not carry out the proof here since apart from technicalities it does not involve any other ideas than those used to prove Theorems 5.4.1 and 6.1.1. Instead we refer the reader to Lax [3].