Pointwise decay for the wave equation on nonstationary spacetimes

Abstract

This first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak form of the integrated local energy decay estimate and proves that solutions to the linear wave equation with variable coefficients in R1+3, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of these functions decay at spatial infinity. Roughly speaking, given metric perturbation terms and first-order terms whose vector fields decay at least as rapidly as the rate r−1−a, and potentials whose vector fields decay at least as rapidly as the rate r−2−a, our results imply that |ZJϕ(t,x)|≤Ct−2−a for bounded |x|, where Z denotes vector fields and J is a multi-index. We expect that this is a sharp decay rate. We prove results for both stationary and nonstationary metrics and coefficients. The proof uses integrated local energy decay to achieve an initial decay rate, and then uses an iteration involving the one-dimensional reduction (due to the positivity of the fundamental solution of the wave equation in three spatial dimensions) to achieve the full decay rate. For second-order perturbations of the Minkowski metric with coefficients that are not necessarily spherically symmetric, we analyze these coefficients' decay rates in a novel way to obtain a higher rate of pointwise decay for the solution than might be expected, assuming only weak pointwise decay on up to two time derivatives of the coefficients (and vector fields thereof); the latter weak decay is satisfied if, for example, the metric is stationary.