Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911231272550400 |
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| author | Luk, Jonathan Oh, Sung-Jin Yu, Dongxiao |
| author_facet | Luk, Jonathan Oh, Sung-Jin Yu, Dongxiao |
| contents | We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global asymptotics towards the future. We prove that the late-time asymptotics is given by a continuous superposition of decay rates, in stark contrast to equations satisfying a null condition.
The asymptotic formula we obtain is given in terms of a solution to the linear wave equation. Combining this with analysis on the linear wave equation, we strengthen some rigidity results of the third author, showing in particular that any solution with a faster time decay than expected away from the wave zone must vanish identically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition Luk, Jonathan Oh, Sung-Jin Yu, Dongxiao Analysis of PDEs We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global asymptotics towards the future. We prove that the late-time asymptotics is given by a continuous superposition of decay rates, in stark contrast to equations satisfying a null condition. The asymptotic formula we obtain is given in terms of a solution to the linear wave equation. Combining this with analysis on the linear wave equation, we strengthen some rigidity results of the third author, showing in particular that any solution with a faster time decay than expected away from the wave zone must vanish identically. |
| title | Late-time tail for a scalar quasilinear wave equation satisfying the weak null condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.22122 |