Pointwise decay for the energy-critical nonlinear wave equation

Abstract: This second article in a two-part series (following [arXiv:2105.02865], listed here as \cite{L}) proves optimal pointwise decay rates for the quintic defocusing wave equation with large initial data on nonstationary spacetimes, and both the quintic defocusing and quintic focusing wave equations with small initial data on nonstationary spacetimes. We prove a weighted local energy decay estimate, and use local energy decay and Strichartz estimates on these variable-coefficient backgrounds. By using an iteration scheme, we obtain the optimal pointwise bounds. In addition, we explain how the iteration scheme reaches analogous pointwise bounds for other integral power nonlinearities that are either higher or lower than the quintic power, given the assumption of global existence for those powers (and in the case of the lower powers, given certain initial decay rates).