Almost global existence for solutions of a nonlinear fourth order hyperbolic equation

We study the long time existence of the solutions of the Cauchy problem for a class of nonlinear fourth order hyperbolic equations, in which the principal part is given by the composition of two waves operators with different propagation speeds. The presence of these two speeds makes this problem essentially different from the correspondent one for the second order wave equation. In fact, in the present case, we cannot apply the approach of S. Klainerman, based on the invariance of the D'Alembertian operator under the complete Lorentz group. Furthermore, even the alternative method used by F. John and S. Klainerman, only for the wave equation in three space dimensions, does not seem directly applicable. Nevertheless, thanks to the special structure of the equations, we are able to show the almost global existence of the solutions to the related Cauchy problem. Received December 29, 1995