Global existence and pointwise decay for nonlinear waves under the null condition

TL;DR

This paper establishes global existence and optimal decay rates for solutions to nonlinear wave equations satisfying the null condition on certain non-stationary, asymptotically flat spacetimes, extending classical results to more general settings.

Contribution

It proves global existence and sharp decay estimates for nonlinear wave equations under a generalized null condition on non-stationary backgrounds, without requiring the null structure to be preserved under commutation.

Findings

Solutions decay as |φ(t,x)| ≲ ⟨t+r⟩^{-1}⟨t−r⟩^{-1}

Global existence holds for small initial data under local energy decay assumptions

Decay rates match those on flat Minkowski space, even with time-dependent perturbations

Abstract

This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes. We consider nonlinearities satisfying a generalized null condition which does not necessarily retain its structure when commuted with vector fields. For sufficiently small initial data, and under the assumption that the underlying linear operator satisfies an integrated local energy decay estimate, we prove that solutions exist for all time and we establish sharp pointwise decay estimates for the solution $\phi$ and its vector-fields. The solution itself decays as $|\phi(t,x)| \lesssim \langle t+r \rangle^{-1} \langle t-r \rangle^{-1}$. This rate matches that of the nonlinear equation on a flat background. This rate is sharp, as this behavior holds already for certain time-dependent perturbations of the classical null form on Minkowski space, which we specify.