Appears in collection : Partial Differential Equations, Analysis and Geometry

I will present a joint work with Sung-Jin Oh (Berkeley), where we develop a general method for understanding the precise late-time asymptotic behavior of solutions to linear and nonlinear wave equations in odd spatial dimensions. In particular, we prove that in the presence of a nonlinearity and/or a dynamical background, the late time tails are in general different from the better understood case of linear equations on stationary backgrounds. I will explain how the late time tails are related to the problem of the singularity structure in the interior of generic dynamical vacuum black holes in general relativity.