Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity
The author considers the scattering problem for the nonlinear Klein-Gordon equation in space-time \(\mathbb R\times \mathbb R^n\) of the form \[ \partial_t^2u-\triangle u+u=F_{\gamma }(u), \] where \(u\) is a real valued or a complex-valued unknown function of \((t,x)\in \mathbb R\times \mathbb R^n\), \(\partial_t=\partial /\partial t\), \(\triangle \) is the Laplacian in \(\mathbb R^n\). The nonlinearity \(F_{\gamma }(u)\) is a cubic convolution term \(F_{\gamma }(u) = -(V_{\gamma }\ast | u| ^2)u\) with the estimate \(| V_{\gamma }(x)| \leq C| x| ^{-\gamma }\) (\(0<\gamma <n\)). Here \(\ast \) denotes the convolution in the space variables. The term \(F_{\gamma }(u)\) is an approximative expression of the nonlocal interaction of specific elementary particles. This mathematical model has been considered by G. Menzala and W. Strauss (1982). The main result is that there exists a scattering operator for the problem under consideration. The author proves this applying the Strichartz estimates for the inhomogeneous Klein-Gordon equation.
scientific article; zbMATH DE number 5140561