Asymptotic Expansions for Semilinear Waves on Asymptotically Flat Spacetimes

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)φ³, we prove φ(t, x) = 2c₀ t⁻² + O(t⁻³⁺) in compact spatial regions, with c₀ computable. For a(t,x)φᵖ with p ≥ 4, we show φ(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.