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 $p \geq 4$, 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.
From: Sam Looi [view email]
[v1]
Fri, 12 Jul 2024 05:32:21 UTC (55 KB)
[v2]
Mon, 29 Jul 2024 22:31:28 UTC (55 KB)
[v3]
Sun, 21 Dec 2025 03:44:07 UTC (57 KB)
[v4]
Tue, 23 Jun 2026 17:55:24 UTC (57 KB)
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