We consider a second-order weakly hyperbolic equation with time and space depending coefficients. We suppose the coefficients to have globally a H\"older type behavior and locally a blow up of the first derivative at some time. We show that the Cauchy problem for such an equation is well posed in Gevrey classes $G^\sigma$; the upper bound for the Gevrey index $\sigma$ depends only on the dominant between the local and the global condition.
"The cauchy problem for a weakly hyperbolic equation with unbounded and non-Lipschitz-continuous coefficients." Differential Integral Equations 20 (4) 467 - 480, 2007. https://doi.org/10.57262/die/1356039463