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We obtain sharp decay estimates and asymptotics for small solutions to theone-dimensional Klein-Gordon equation with constant coefficient cubic andspatially localized, variable coefficient cubic nonlinearities. Vector-fieldtechniques to deal with the long-range nature of the cubic nonlinearity becomeproblematic in the presence of variable coefficients. We introduce a novelapproach based on pointwise-in-time local decay estimates for the Klein-Gordonpropagator to overcome this impasse.

Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities