Asymptotic Expansions for Semilinear Waves on Asymptotically Flat Spacetimes

Shi-Zhuo Looi, Haoren Xiong

Published 2024-07-12Version 1

We establish precise asymptotic expansions for solutions to semilinear wave equations with power-type nonlinearities on asymptotically flat spacetimes. For cubic nonlinearities $a(t,x)\phi^3$, we prove $\phi(t, x) = 2c_0 t^{-2} + O(t^{-3+})$ in compact spatial regions, with $c_0$ computable. For $a(t,x)\phi^p$ with $p \geq 4$, we show $\phi(t, x) = d t^{-3} + O(t^{-4+})$, extending Price's law to the nonlinear setting. Our approach combines radiation field analysis with a generalized low-energy resolvent expansion, providing a bridge between spectral and physical space methods. These results sharpen previous decay estimates and yield complete asymptotics across the entire spacetime, including black hole backgrounds.