Propagation of analytic and Gevrey regularity for a class of semi-linear weakly hyperbolic equations

The paper deals with the problem of the propagation of analytic regularity of the solutions to the weakly hyperbolic equations $\partial^m_tu+G(t,x,\partial^\alpha_{t,x}u)_{|\alpha|\leq m-1}=0$. The authors prove that the solution is analytic in every cylinder $[0,T]\times \omega$ if $G$ and the initial data are analytic, and the solution $u$ is assumed to be in some Gevrey class of order $\sigma_1$ smaller than $1/\rho$ with an index $\rho \leq 1-1/m$ which is determined by the derivatives of $u$ that appear in $G$