Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes

Abstract

We prove quantitative scattering for the three-dimensional defocusing energy-critical quintic wave equation on a class of asymptotically flat, possibly non-stationary perturbations of Minkowski space, by establishing the first explicit global $L^8_{t,x}$ bound in this variable-coefficient setting. Earlier work in this setting proved scattering only qualitatively. For \[ Pu=u^5,\qquad P=\partial_\alpha(g^{\alpha\beta}\partial_\beta), \] we show that, under smallness, decay, and regularity assumptions on the metric, and assuming a priori $\dot H^5\times\dot H^4$ and $L^2\times\dot H^{-1}$ bounds on the solution, the critical spacetime norm $\|u\|_{L^8_{t,x}(\mathbb R\times\mathbb R^3)}$ satisfies an explicit exponential-type estimate in terms of the energy and the a priori bound. This upgrades the qualitative scattering theory in this setting to a quantitative one. The main difficulty is to control geometric error terms over long times. We handle this by splitting the Duhamel history into recent past and remote past. For the recent past, we prove a variable-coefficient interaction Morawetz estimate that yields, on every sufficiently long interval, a time at which the recent nonlinear contribution is small. For the remote past, we prove a dispersive estimate from integrated local energy decay together with a transfer of pointwise decay from large radius to large time. Combining these estimates gives the explicit global $L^8_{t,x}$ bound.