A sufficient condition for well-posedness for systems with time-dependent coefficients

We consider linear, smooth, hyperbolic systems with time-dependent coefficients and size N. We give a condition sufficient for the well-posedness of the Cauchy Problem in some Gevrey classes. We present some Levi conditions to improve the Gevrey index of well-posedness for the scalar equation of order N, using the transformation in, [DAS] and following the technique introduced in [CT]. By using this result and adding some assumptions on the form of the first-order term, we can improve the well-posedness for systems. A similar condition has been studied in [DAT] for systems with size 3.