https://hal.univ-brest.fr/hal-00470395v2Joudioux, JérémieJérémieJoudiouxLM - Laboratoire de mathématiques de Brest - UBO - Université de Brest - IBNM - Institut Brestois du Numérique et des Mathématiques - UBO - Université de Brest - CNRS - Centre National de la Recherche ScientifiqueConformal scattering for a nonlinear wave equation on a curved backgroundHAL CCSD2010scatteringnonlinear wave equationrelativity[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP][MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph][PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph]Joudioux, Jérémie2010-10-28 11:14:232021-10-11 14:22:072010-10-29 16:11:42enPreprints, Working Papers, ...https://hal.univ-brest.fr/hal-00470395v2/documenthttps://hal.univ-brest.fr/hal-00470395v1application/pdf2The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by Hörmander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator.