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| We consider the Cauchy problem for a class of hyperbolic operators
which satisfy the following conditions: \\
(1) The coefficients depends only on the time variable. \\
(2) The coefficients of the principal parts are real analytic. \\
(3) The multiplicities of the characteristic roots are at most two unless
the operators are of third order. \\
\quad Then we show that the Cauchy problem for the operators is $C^\infty $
well-posed under Levi type conditions. Namely, for $C^\infty $ well-posedness
we impose some conditions on the subprincipal symbols, and, in addition, on
so-called \lq \lq sub-sub-principal symbols'' if the operators are of third
order. |
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