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. For cubic nonlinearities $a(t,x)\phi^3$, we prove $\phi(t, x) = 2c t^{-2} + O(t^{-3+})$ in compact spatial regions, with $c$ computable. For $a(t,x)\phi^p$ with $p \geq 4$, we show $\phi(t, x) = d t^{-3} + O(t^{-4+})$, 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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.08997
- arXiv:
- arXiv:2407.08997
- Bibcode:
- 2024arXiv240708997L
- Keywords:
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- Mathematics - Analysis of PDEs;
- General Relativity and Quantum Cosmology
- E-Print:
- 48 pages