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

We consider the scattering problem for the Klein-Gordon equa-tion with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation. This paper is concerned with the scattering problem for the nonlinear Klein-Gordon equation of the form ∂2t u−∆u + u = Fγ(u) (1) in space-time R×Rn, where u is a real-valued or a complex-valued unknown function of (t, x) ∈ R × Rn, ∂t = ∂/∂t and ∆ is the Laplacian in Rn. The nonlinearity Fγ(u) is a cubic convolution term Fγ(u) = (Vγ ∗ |u|2)u with |Vγ(x) | ≤ C|x|−γ. (2) Here, 0 < γ < n and ∗ denotes the convolution in the space variables. Th