Levi conditions and global Gevrey regularity for the solutions of quasilinear weakly hyperbolic equations

In the present paper the authors prove a global Gevrey regularity result for the solutions of the Cauchy problem for the quasilinear weakly hyperbolic equation with spatial degeneracy u(tt) - (a(x, t)u(x))(x) = f(x,t, u, u(z)). The basic tool is a well-posedness result (local existence and cone of dependence) in Gevrey spaces. Such a result can be proved only under the assumption of Levi conditions. Suitable energy estimates lead to the regularity of solutions. This result generalizes results from the strictly hyperbolic case.