Small data scattering for the Klein-Gordon equation with a cubic convolution

Abstract

We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We present the method to prove the existence of the scattering operator on a neighborhood of 0 in the weighted Sobolev space $H^{s,\sigma} = (1- \Delta)^{-s/2}\ \langle x \rangle^{-\sigma}\ L_2 (\mathbb{R}^n)$. The method is based on the complex interpolation method of the weighted Sobolev spaces and the Strichartz estimates for the inhomogeneous Klein-Gordon equation.