Propagation of analyticity for a class of nonlinear hyperbolic equations

Abstract

We consider the semilinear hyperbolic equations of the form

with $f(u)$ entire analytic, where the characteristic roots satisfy

We prove that, if the $a_h(t)$'s are analytic functions, al the solutions bounded in ${\mathcal{C}}^{\infty }$ enjoy the propagation of analyticity; while, if the $a_h(t)$'s are ${\mathcal{C}}^{\infty }$ functions, such a property holds for those solutions which are bounded in some Gevrey class.