On long time stability in Hamiltonian perturbations of non-resonant linearPDEs

We consider small Hamiltonian perturbations of a system of infinitely many harmonic oscillators. We assume that the frequencies lambda(j), j greater t han or equal to 1, fulfil lambda(j) similar to j(d) with d greater than or equal to 1, and a suitable non-resonance condition of Diophantine-type. We prove a Nekhoroshev-type theorem for solutions corresponding to initial dat a such that the energy on the jth linear oscillator is bounded by Ke(-gamma ja) with a given a > 1 and positive gamma, K. We show, precisely that up t o times of order exp \1n epsilon\(1+b) where epsilon is the size of the per turbation and b a positive parameter, the solution remains close to a quasi -periodic motion. Closedness is measured in a weighted e(2) norm with an ex ponential weight. For our Diophantine-type condition we show that if the fr equencies depend on a real parameter, and a suitable non-degeneracy conditi on is fulfilled, it is satisfied for almost all values of the parameter. Fi nally, we apply the general result to some nonlinear Klein-Gordon equations .