The cauchy problem for a weakly hyperbolic equation with unbounded and non-Lipschitz-continuous coefficients

Abstract

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.