A scattering construction for nonlinear wave equations on Kerr--Anti-de Sitter spacetimes

Abstract

Existence of a large class of exponentially decaying solutions of the nonlinear massive wave equation $\Box_g\psi+\alpha\psi = \mathcal{F}(\psi,\partial\psi)$ on a Kerr--Anti-de Sitter exterior is established via a backwards scattering construction. Exponentially decaying data is prescribed on the future event horizon, and Dirichlet data on the timelike conformal boundary. The corresponding solutions exhibit the full functional degrees of freedom of the problem, but are exceptional in the sense that (even) general solutions of the forward, linear ($\mathcal{F}=0$) problem are known to decay at best inverse logarithmically. Our construction even applies outside of the \textit{Hawking-Reall bound} on the spacetime angular momentum, in which case, there exist exponentially growing mode solutions of the forward problem. As for the analogous construction in the asymptotically flat case, the assumed exponential decay of the scattering data on the event horizon is exploited to overcome the gravitational blueshift encountered in the backwards construction.