Research on Systems of Partial Differential Equations Appling the Abstract Algebra

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We have three themes. The first is to establish the necessary and sufficient condition for the Cauchy-Kowalevskaya theorem for systems of partial differential equations including the generalization to the, Nagumo type. On the original Cauchy-Kowalevskaya theorem, we obtained clearer proof. On the theorem of Nagumo type, we obtained a proof on the necessity and also a proof on the sufficiency in a special case. The second is the characterization of the strong hyperbolicity on systems. On the example which is pointwisely diagonalizable but not strongly hyperbolic by Petrovsky, we have already known that it changes to a strongly hyperbolic system by any hyperbolic perturbation. We showed that this phenomenon occurs generally for the systems with time-dependent coefficients. To show this, we apply the solvability of the Cauchy problem for the systems of Fuchs type. We also succeeded the generalization of the structure of the solvability. The third is the solvability of the Cauchy problem for p-parabolic systems to the future. Unfortunately, we obtained an idea to solve this problem, but finally we cannot achieve it as the complete form. In these researches, the comparison between the calculation on the non-commutative ring of the meromorphic formal symbols and that on the holomorphic pseude-differential operators has played an essential role.At first, the Kac problem is not the theme of this project. We obtained a good knowledge on the non-commutative groups through the research on the determinatnt theory on non-commutative ring and it brings a viewpoint on the framework of the existence of the counterexamples on the Kac problem by Sunada. As a result, we obtained concrete example of the domains which change from convex to nonconvex smoothly and for which the Kac problem is affirmative by proving mathematically Watanabe's conjecture by the numerical try. This is the first offer of a nonconvex domain for which the Kac problem is affirmative.