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Vanishing of the Fefferman-Graham obstruction tensor was used by Anderssonand Chru{\'s}ciel to show stability of the asymptotically de Sitter spaces ineven dimensions. However, existing proofs of hyperbolicity of this equationcontain gaps. We show in this paper that it is indeed a well-posed hyperbolicsystem with unique up to diffeomorphism and conformal transformations smoothdevelopment for smooth Cauchy data. Our method applies also to equationsdefined by various versions Graham-Jenne-Mason-Sparling operators. Inparticular, we use one of these operators to propagate Gover's condition ofbeing almost conformally Einstein. This allows to study initial data also forCauchy surfaces which cross conformal boundary. As a by-product we show that onglobally hyperbolic manifolds one can always choose conformal factor such thatBranson Q-curvature vanishes.

Well-posedness of the ambient metric equations and stability of even dimensional asymptotically de Sitter spacetimes