Wave and scattering operators for the nonlinear Klein-Gordon equation on a quarter-plane

Abstract

We consider the initial-boundary value problem for the nonlinear Klein-Gordon equation posed on a quarter-plane with Dirichlet non zero boundary conditions. We construct the wave and scattering operators for this model in the scattering-supercritical regime. We study the influence of the boundary data on the asymptotic behavior of solutions. We show that if the boundary data has sufficiently slow decay rate, it affects the order of the nonlinearity for which we are able to construct the scattering operator. This suggests a modification for the critical value of the nonlinear term due to the presence of non zero boundary condition. In order to prove our results, we construct an explicit formula for the solutions to corresponding linear Klein-Gordon equation on a quarter-plane. Using this explicit formula, we obtain weighted Sobolev estimates for the solution to the complete nonlinear problem. In order to control the long-time behavior of the solution we find an operator J suitable for our settings, analogous to operator x+it∂x in the case of NLS equation.