Abstract
Abstract Let P(x, t, D I , D t ) be a formally hyperbolic differential operator with constant multiplicity. This article gives a brief survey of the development of the research on the Cauchy problem for P. In the C ∞ -case, the necessary and sufficient condition for the C ∞ wellposedness (Levi condition) is given in an explicit form by using the perfect factorization of P. Next, when the Levi condition is violated, the wellposedness in the spaces of Gevrey class is discussed. The best possible class of Gevrey is given also through the perfect factorization of P
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