A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

Abstract

NON-FUCHSIAN PARTIAL DIFFERENTIAL OPERATORS TAKESHI MANDAI §1. IntroductionConsider a partial differential operatorwhere K is a non-negative integer and a ia are real-analytic in a neighborhood of (0,0) €= R, x R* M. S. Baouendi and C. Goulaouic [1] defined Fuchsian partial differential operators, and proved the unique solvability of the characteristic Cauchy problems in the category of real-analytic (or holomorphic) functions, which is a generalization of the classical Cauchy-Kowalevsky theorem. They also proved a generalization of the Holmgren uniqueness theorem. Especially, from their results it easily follows that if P is a Fuchsian operator with real-analytic coefficients, then there exist no sufficiently smooth null-solutions. Here, a Schwartz distribution u in a neighborhood of (0,0) is called a null-solution for P at (0,0), if Pu = 0 in a neighborhood of (0,0) and (0,0) e supp u c {t > 0}, where supp u denotes the support of u.The author considered the characteristic Cauchy problems for a class of operators wider than the Fuchsian operators in [3]. In that result, he showed the unique solvability of the characteristic Cauchy problems in the category of functions which are of class C°° with respect to t and real-analytic with respect to x. He also showed the non-existence of sufficiently smooth null-solutions. (As for Receive April 28, 1995.