Dynamics of Infinite Classical Anharmonic Crystals

We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential V and is subjected to a restoring force of potential U . We assume that U and V are even nonnegative polynomials of degree 2σ1 and 2σ2. We study the time evolution of this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case σ1 < 2σ2 − 1 of some already known results obtained for σ1 ≥ 2σ2 − 1.