Asymptotic expansions for semilinear waves on 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 $$a(t,x) \, \phi ^3$$ a ( t , x ) ϕ 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 $$\phi (t, x) = c \, t^{-2} + \mathcal {O}(t^{-3+})$$ ϕ ( t , x ) = c t - 2 + O ( t - 3 + ) . For higher-order nonlinearities $$a(t,x) \, \phi ^p$$ a ( t , x ) ϕ p with $$p \ge 4$$ p ≥ 4 , we prove the solution satisfies $$\phi (t, x) = d \, t^{-3} + \mathcal {O}(t^{-4+})$$ ϕ ( 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.