Stability and Instability of Schwarzschild-AdS for the Nonlinear Einstein-Klein-Gordon System

In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled Nonlinear-Einstein-Klein-Gordon (NLEKG) system in the presence of a negative cosmological constant. We prove the well posedness of the NLEKG system in the Schwarzschild-AdS spacetimes and that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which φ = 0 identically) are asymptotically stable. Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a Schwarzschild-AdS spacetime. Bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.