The Cauchy Problem for Hyperbolic Systems with Variable Multiple Characteristics

Abstract

Recently the Cauchy problem for weakly hyperbolic operators has been studied by many authors. In case of equations with constant multiple characteristics of multiplicity 2, Mizohata and Ohya [7] gave a necessary and sufficient condition for well-posedness of the problem. For the case of higher multiplicity the problem was solved by Flashka-Strange [8] and Chazarain [9] by introducing a generalized Levi's condition. For the case where variable multiple characteristics are concerned, Oleinik [3] obtained a sufficient condition for well-posedness to the equations of the second order. Further, Menikoff [2] extended Oleinik's results to the equations of higher order. Then Ohya [1] simplified Menikoff's proof through extending the method in [7] in a natural manner. For weakly hyperbolic systems with constant multiple characteristics, either necessary or sufficient conditions for well-posedness are given by Petkov [4], Yamahara [5] and others (c.f. Demay [6]). In this paper, we shall give a sufficient condition for the Cauchy problem to be well-posed to a first order hyperbolic system with variable multiple characteristics. The proof is done by a method along the ideas of [6] combined with the method of [1]. I should like to express my sincere thanks to Professor S. Irie and Professor T. Kakita for their valuable suggestions and kind encouragement.