Modified scattering for the mixed initial-boundary problem for the nonlinear Klein?Gordon equation

In the present paper we consider the mixed initial-boundary value problem for the cubic nonlinear Klein?Gordon equation with non zero boundary data. The main purpose is to investigate the existence of modified scattering for this model. We obtain sharp decay estimates in the uniform norm for the solutions as well as a sharp asymptotic profile, which has a modified character due to the gauge invariant cubic nonlinearity. The proof is based on the solution to the corresponding linear problem, a factorization technique and the implementation of a conformal-type operator. We find a decomposition for the nonlinearity, which allows us to separate the resonant part and to calculate the modification in the scattering of the solutions due to this resonance.