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. Our analysis focuses on two key cases: cubic nonlinearities and higher-order power nonlinearities. For cubic nonlinearities of the form $a(t,x)ϕ^3$, we prove asymptotic expansions for the solution globally in the spacetime. In the special case of compact spatial regions, solutions exhibit the asymptotic behavior $ϕ(t,x) = ct^{-2} + O(t^{-3+})$. For higher-order nonlinearities $a(t,x)ϕ^p$ with $p\geq 4$, we prove the solution satisfies $ϕ(t, x)= d t^{-3} + O(t^{-4+})$, 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.

💡 Research Summary

The paper establishes precise late‑time asymptotic expansions for solutions of semilinear wave equations on stationary, asymptotically flat spacetimes, extending the classical linear Price law to nonlinear settings. The authors consider the initial value problem
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