The Cauchy Problem for Quasilinear Hyperbolic Equations with Non-Absolutely Continuous Coefficients in the Time Variable

We consider the Cauchy problem for a quasilinear weakly hyperbolic operator with coefficients having the first time derivative with singular behavior of the type t^{−q}, q > 1, as t → 0. We show that for t ≤ T_0 , T_0 sufficiently small, given Cauchy data in a Gevrey class there exists a unique solution u in Gevrey classes of order σ < (qr)/(qr−1) where r denotes the largest multiplicity of the characteristic roots