Local Energy control in the presence of a zero-energy resonance

We consider the problem of stability and local energy decay for co-dimension one perturbations of the soliton of the cubic Klein-Gordon equation in $1+1$ dimensions. Our main result gives a weighted time-averaged control of the local energy over a time interval which is exponentially long in the size of the initial (total) energy. More precisely, for well-prepared initial perturbations on the center stable manifold that are of size $\delta$ in the energy norm, we show that the local energy is under control up to times of the order $\exp(c\delta^{-\beta})$ for any $\beta < 4/3$. A major difficulty is the presence of a zero-energy resonance in the linearized operator, which is a well-known obstruction to improved local decay properties. We address this issue by using virial estimates that are frequency-localized in a time-dependent way and introducing a "singular virial functional" with time-dependent weights to control the mass of the perturbation projected away from small frequencies. The proof applies to more general models, yielding analogous results for perturbations of the kink of the Sine-Gordon model, and small solutions of nonlinear Klein-Gordon equations. In this respect, our result is close to optimal due to the existence of wobbling kinks and breathers in the Sine-Gordon model which violates our conclusion if $\beta = 2$. This appears to be the first successful general attempt at using virial estimates in the presence of a resonance to deduce local energy control.