By means of a backwards scattering construction, existence of a class of exponentially decaying solutions of the nonlinear massive wave equation $\Box_g\psi+\alpha\psi = \mathcal{F}(\psi,\partial\psi)$ on the Kerr-Anti-de Sitter black hole exterior region is proven. Scattering data characterising the exponential decay is posed on the future event horizon, paired with Dirichlet boundary conditions at null infinity. The resulting solutions form a large class, exhibiting all functional degrees of freedom associated with the problem. However, given that (at best) inverse logarithmic decay has been established for even general solutions of the linear ($\mathcal{F}=0$) forward problem \cite{SharpDecay}, they are decidedly non-generic. The \textit{Hawking-Reall bound} on the angular momentum of the spacetime is not imposed. In this setting, it is known that the forward problem admits exponentially growing mode solutions \cite{Dold}. The prescribed exponential decay of the horizon data is utilised in overcoming the \textit{blueshift effect} one encounters when solving backwards, as is familiar from the related result in the asymptotically flat setting \cite{SchwzScattering}.

In order to establish key techniques for the scattering construction on the Kerr-Anti-de Sitter exterior, similar constructions on the asymptotically flat Minkowski space and Schwarzschild black hole exterior are presented. These are followed by a nonlinear local existence result on pure Anti-de Sitter space, serving as an introduction to asymptotically Anti-de Sitter spacetimes and nonlinear techniques.