Asymptotic expansions for semilinear waves on asymptotically flat spacetimes

TL;DR

This paper derives precise asymptotic expansions for solutions to semilinear wave equations on asymptotically flat spacetimes, extending classical decay laws to nonlinear cases using advanced microlocal analysis techniques.

Contribution

It introduces a novel approach combining geometric microlocal analysis with physical-space methods to analyze nonlinear wave asymptotics, extending Price's law.

Findings

Asymptotic expansion for cubic nonlinearities: $\,ct^{-2} + O(t^{-3+})$

Asymptotic behavior for higher-order nonlinearities: $\,dt^{-3} + O(t^{-4+})$

Extension of Price's law to nonlinear wave equations in asymptotically flat spacetimes.

Abstract

We establish precise asymptotic expansions for solutions to semilinear wave equations with power-type nonlinearities on asymptotically flat spacetimes. Our analysis focuses on two key cases: cubic nonlinearities and higher-order power nonlinearities. For cubic nonlinearities of the form $a(t,x)\phi^3$, we prove asymptotic expansions for the solution globally in the spacetime. In the special case of compact spatial regions, solutions exhibit the asymptotic behavior $\phi(t,x) = ct^{-2} + O(t^{-3+})$. For higher-order nonlinearities $a(t,x)\phi^p$ with $p\geq 4$, we prove the solution satisfies $\phi(t, x)= d t^{-3} + O(t^{-4+})$, thereby extending the classical Price's law (a late-time tail postulated in 1972) to nonlinear settings in a precise fashion. These results sharpen previous decay estimates for nonlinear waves. We develop a radiation field expansion and a low-energy resolvent expansion adapted to conormal asymptotic inputs, extending Hintz's approach for linear waves to the semilinear setting. Our methods connect geometric microlocal analysis (b-calculus) with classical physical-space techniques, providing a convenient tool for analyzing asymptotic behavior of nonlinear waves.