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.
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