Theory of hyperloobic systems

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We have obtained a lot of results. We refer here some of the main results.1. For 2 x 2 systems with two independent variables, we obtained a necessary and sufficient condition in order that the Cauchy problem is well posed. The condition is expressed using the Newton polyhedron. In this study we found a peculiar example which is strictly hyperbolic apart from the initial plane for that the Cauchy problem is not well posed for any lower order term.2. We introduced a new notion "pseudo-symmetric hyperbolic systems" which extends the symmetrizable hyperbolic systems. We proved that the Cauchy problem for pseudo-symmetric hyperbolic systems with one space variable is well posed. The question is still open for pseudo-symmetric systems with several space variables.3. We succeeded in obtaining a necessary condition on lower order terms for the Cauchy problem is well posed for general, hyperbolic systems using the determinant on a non commutative field where the localization lives : the. leading part of the non commutative determinant of the localization of the total symbol coincides with the principal part of the classical determinant of the principal symbol.4. The symmetrizability of the frozen system at every space point implies the symmetrizability of the original systm if the reduced dimension is enough high. In particular if the every frozen system is stringly hyperbolic then the original system is also strongly hyperbolic if the reduced dimension is high.