Global existence in infinite lattices of nonlinear oscillators: The Discrete Klein-Gordon equation

Abstract

Pointing out the difference between the Discrete Nonlinear Schr\"odinger equation with the classical power law nonlinearity-for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice-we prove either global existence or nonexistence in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of ``blow-up'' sign), under suitable conditions on the initial data, and some times on the dimension of the lattice. The results, consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts, are remarked. We also make a short comment, on the existence of excitation thresholds, for forced solutions of damped and parametrically driven, Klein-Gordon lattices.