I will discuss recent results on the quartic nonlinear Klein-Gordon equation with a potential in one dimension, motivated by the stability of the kink in the $\phi^4$ model. Assuming the presence of an internal mode (a positive gap eigenvalue), which obstructs decay at the linear level, we construct small global solutions that decompose into an internal mode and dispersive radiation, and show that the internal mode decays via the Fermi Golden Rule.

Some of the main challenges arise from slow dispersive decay in 1D, resonant interactions between the discrete and continuous spectra, and the low-power nonlinearity. To overcome these difficulties, we introduce new ideas combining distorted Fourier analysis, normal form transformations, and refined dispersive and smoothing estimates.