Non-uniqueness and uniqueness in the Cauchy problem at simple characteristic points

Abstract

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.