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

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.