We establish a geometric scattering theory for a conformally invariant nonlinear wave equation on an asymptotically simple space-time. The scattering operator is defined via some trace operators at null infinity, and the proof is decomposed into three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field to the Schwarzschild space-time and a method introduced by Hörmander for the Goursat problem. A well-posedness theorem for the characteristic Cauchy problem on a light cone at infinity is then obtained. Its proof requires a control of the nonlinearity that is uniform in time and follows from, both, an estimate of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinity are introduced and allow us to define the conformal scattering operator of interest.
Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity
