REMARKS ON THE ENERGY SCATTERING FOR NONLINEAR KLEIN-GORDON AND SCHRÖDINGER EQUATIONS

Abstract

We give an improved proof for the result established recently by the present author that the scattering operators are well-defined in the whole energy space for a class of nonlinear Klein-Gordon and Schrödinger equations in any spatial dimension. Using some Sobolev-type inequatilies, we can simplify and somewhat enhance the Morawetz-type estimates and thereby weaken the required repulsivity conditions.