On long time stability in Hamiltonian perturbations of non-resonant linear PDEs

We consider small Hamiltonian perturbations of a system of infinitely many harmonic oscillators. We assume that the frequencies j , j 1, fulfil j ~ j d with d 1, and a suitable non-resonance condition of Diophantine-type. We prove a Nekhoroshev-type theorem for solutions corresponding to initial data such that the energy on the j th linear oscillator is bounded by K exp(- j a ) with a given a > 1 and positive , K . We show, precisely that up to times of order exp|ln | 1+ b , where is the size of the perturbation and b a positive parameter, the solution remains close to a quasi-periodic motion. Closedness is measured in a weighted 2 norm with an exponential weight. For our Diophantine-type condition we show that if the frequencies depend on a real parameter, and a suitable non-degeneracy condition is fulfilled, it is satisfied for almost all values of the parameter. Finally, we apply the general result to some nonlinear Klein-Gordon equations.