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In this paper, we study the global behavior of solutions to the sphericallysymmetric coupled Einstein-Klein-Gordon (EKG) system in the presence of anegative cosmological constant. We prove that the Schwarzschild-AdS spacetimes(the trivial black hole solutions of the EKG system for which $\phi=0$identically) are asymptotically stable: Small perturbations ofSchwarzschild-AdS initial data again lead to regular black holes, with themetric on the black hole exterior approaching a Schwarzschild-AdS spacetime.The main difficulties in the proof arise from the lack of monotonicity for theHawking mass and the asymptotically AdS boundary conditions, which render even(part of) the orbital stability intricate. These issues are resolved in abootstrap argument on the black hole exterior, with the redshift effect andweighted Hardy inequalities playing the fundamental role in the analysis. Bothintegrated decay and pointwise decay estimates are obtained.

Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system