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We study the 1D Klein-Gordon equation with variable coefficient nonlinearity.This problem exhibits an interesting resonant interaction between the spatialfrequencies of the nonlinear coefficients and the temporal oscillations of thesolutions. In the case when only the cubic coefficients are variable we provedispersive decay and smoothness of the solution in weighted spaces with thehelp of quadratic and cubic normal-forms transformations. In the case of cubicinteractions these normal forms appear to be novel.

Dispersive Decay for the 1D Klein-Gordon Equation with Variable Coefficient Nonlinearities