Well posedness under Levi conditions for a degenerate second order Cauchy problem

We consider the Cauchy problem for a second order equation of hyperbolic type which degenerates both in the sense that it is weakly hyperbolic and it has non Lipschitz continuous in time coefficients. Intersections between the roots of the equation are of a finite order k, and the first time derivative of the principal part's coefficients present a blow-up phenomenon at the time t=0, behaving as t^{-q}, q greater or equal 1. The mixture of these two situations gives, under an appropriate Levi condition, C^\infty or Gevrey well posedness of the Cauchy problem, depending on the dominant between the two behaviors.