## Details

Abstract: We will start with a review of previous results concerning instabilities of rotating Kerr black holes to massive scalar field perturbations. These instabilities are connected to superradiance and hence do not occur on the Schwarzschild black hole; one naturally wonders if Schwarzschild could thus play a privileged role in the long time evolution of the Einstein-Klein-Gordon system. As a first step to addressing this, we will discuss recent work where we show that solutions to the corresponding Klein-Gordon equation arising from strongly localized initial data fall off with polynomial rates in time. A key challenge to overcome is the presence of stable timelike geodesics orbiting the black hole which threaten to trap the massive scalar field and lead to slow logarithmic type decay. The talk is based on joint work(s) with Federico Pasqualotto and Maxime Van de Moortel.