Asymptotic expansions for semilinear waves on asymptotically flat spacetimes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909971309920256 |
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| author | Looi, Shi-Zhuo Xiong, Haoren |
| author_facet | Looi, Shi-Zhuo Xiong, Haoren |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions for semilinear waves on asymptotically flat spacetimes Looi, Shi-Zhuo Xiong, Haoren Analysis of PDEs General Relativity and Quantum Cosmology 83C30, 35L71, 47A10, 42A38 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. |
| title | Asymptotic expansions for semilinear waves on asymptotically flat spacetimes |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology 83C30, 35L71, 47A10, 42A38 |
| url | https://arxiv.org/abs/2407.08997 |