Coefficients with unbounded derivatives in hyperbolic equations

We are concerned with the problem of determining the sharp regularity of the coefficients with respect to the time variable $t$ in order to have a well posed Cauchy problem in $H^infty$ or in Gevrey classes for linear or quasilinear hyperbolic operators of higher order. We use and mix two different scales of regularity of global and local type: the modulus of H"older continuity and/or the behaviour with respect to $|t-t_1|^{-q}, qgeq 1,$ of the first derivative as $t$ tends to a point $t_1$. Both are ways to weaken the Lipschitz regularity.