On the Propagation of Singularities of the Solution of the Cauchy Problem

Abstract

In this paper we consider the Cauchy problem for linear partial differential equation with holomorphic coefficients in complex domain. We treat the case where the initial surface is non-characteristic and the initial data have singularities along a regular surface on the initial surface. In Q2j we showed that, under the assumption that the characteristic surfaces issuing from the points of singularity in the initial data are simple and do not touch one another, the singularity in the initial data is propagated along these surfaces and if the initial data have at most poles (resp. essential singularities), the solution in general has at most poles (resp. essential singularities) and logarithmic singularities on these surfaces. However, if an equation admits multiple characteristics, the behavior of the solution is different from that of the case of simple characteristics. For example we consider the Cauchy problem