Remarks on decay of small solutions to systems of Klein–Gordon equations with dissipative nonlinearities

We consider the Cauchy problem for systems of cubic nonlinear Klein–Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution gains an additional logarithmic decay in comparison with the free evolution in the sense of LpLp, 2≤p≤∞2≤p≤∞.