The scattering problem for the three-dimensional cubic nonlinear Klein-Gordon equation with rapidly decreasing input data

Abstract

The scattering problem for the three-dimensional cubic nonlinear Klein-Gordon equation is studied. It has been shown that the scattering operator S is well-defined on a neighborhood in the critical space H1/2(R3)⊕H−1/2(R3) of 0. We prove that if functions f− and g− are in the Schwartz space S(R3) and small in the sense of the critical space, then the corresponding output data (f+,g+):=S(f−,g−) also belongs to S(R3)⊕S(R3). Furthermore, we give sufficient conditions for (f−,g−) such that all order partial derivatives of f+ and g+ decay more rapidly than a same exponential function.