The Cauchy problem for properly hyperbolic equations in one space variable

this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots tau(j) verify an inequality like tau(2)(j) (x) + tau(2 )(k)(x) <= M (tau(j) (x) - tau(k) (x))(2). We prove that the Cauchy problem is well-posed in C-infinity if the operators with frozen coefficients are uniformly hyperbolic in the sense of Garding.