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 \(a(t,x) \, \phi ^3\), we prove asymptotic expansions for the solution globally in the spacetime. In the special case of compact spatial regions, solutions exhibit the asymptotic behavior \(\phi (t, x) = c \, t^{-2} + \mathcal {O}(t^{-3+})\). For higher-order nonlinearities \(a(t,x) \, \phi ^p\) with \(p \ge 4\), we prove the solution satisfies \(\phi (t, x) = d \, t^{-3} + \mathcal {O}(t^{-4+})\), 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.