Time decay for the nonlinear Klein-Gordon equation

It is shown that solutions of the nonlinear Klein-Gordon equation u tt - ∆ u + mu + P '( u ) = 0 decay to zero in the local L 2 mean if the initial energy is bounded provided s P ') s ) - 2 P ( s ) ≥ a P ( s ) ≥ 0 with a > 0. The local energy also decays. The proof is based on manipulating energy identities and re­quires that u have continuouś first derivatives and piecewise continuous second derivatives. The proof is also applicable to certain systems of equations.