Decay rates for cubic and higher order nonlinear wave equations on asymptotically flat spacetimes

TL;DR

This paper establishes pointwise decay rates for cubic and higher order nonlinear wave equations on asymptotically flat spacetimes, including quasilinear cases, under weaker energy decay assumptions, with improved rates for certain nonlinearities.

Contribution

It proves decay rates for nonlinear wave equations on asymptotically flat spacetimes using weaker energy assumptions and improves decay for derivatives-based nonlinearities.

Findings

Proved pointwise decay rates for nonlinear wave equations.

Established decay under weaker energy estimates.

Achieved better decay rates for derivative nonlinearities.

Abstract

In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear equation (rather than the nonlinear equation) satisfies a weaker form of the standard integrated local energy decay, or Morawetz, estimate. For nonlinearities with a total derivative structure, we prove better pointwise decay rates.