  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.
