Time Decay for the Nonlinear Klein-Gordon Equation

It is shown that solutions of the nonlinear Klein-Gordon equation u_tt-4Δ u+mu+P'(u)=0 decay to zero in the local L^2 mean if the initial energy is bounded provided sP'(s)-2P(s) >=slant aP(s) >=slant 0 with a > 0. The local energy also decays. The proof is based on manipulating energy identities and requires that u have continuous first derivatives and piecewise continuous second derivatives. The proof is also applicable to certain systems of equations.