  We obtain sharp decay estimates and asymptotics for small solutions to the
one-dimensional Klein-Gordon equation with constant coefficient cubic and
spatially localized, variable coefficient cubic nonlinearities. Vector-field
techniques to deal with the long-range nature of the cubic nonlinearity become
problematic in the presence of variable coefficients. We introduce a novel
approach based on pointwise-in-time local decay estimates for the Klein-Gordon
propagator to overcome this impasse.
