Pointwise decay for the energy-critical nonlinear wave equation

TL;DR

This paper establishes optimal pointwise decay rates for solutions to the energy-critical nonlinear wave equation on nonstationary spacetimes, extending decay results to large and small initial data for both focusing and defocusing cases.

Contribution

It introduces an iteration scheme combined with local energy decay and Strichartz estimates to achieve optimal decay bounds for the energy-critical wave equation on dynamic backgrounds.

Findings

Proves optimal pointwise decay rates for large initial data in the defocusing case.

Establishes decay bounds for small initial data in both focusing and defocusing cases.

Develops an iteration scheme applicable to various nonlinear powers under global existence assumptions.

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).