Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918085556961280 |
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| author | Yu, Dongxiao |
| author_facet | Yu, Dongxiao |
| contents | We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution $u$ to the scalar wave equation with sufficiently small $C_c^\infty$ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for $|x|<t$). It involves the scattering data obtained in the author's asymptotic completeness result in arXiv:2105.11573. Using this asymptotic formula, we prove that $u$ must vanish under some decaying assumptions on $u$ or its scattering data, provided that the wave equation violates the null condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition Yu, Dongxiao Analysis of PDEs We study the timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition. Given a global solution $u$ to the scalar wave equation with sufficiently small $C_c^\infty$ initial data, we derive an asymptotic formula for this global solution inside the light cone (i.e. for $|x|<t$). It involves the scattering data obtained in the author's asymptotic completeness result in arXiv:2105.11573. Using this asymptotic formula, we prove that $u$ must vanish under some decaying assumptions on $u$ or its scattering data, provided that the wave equation violates the null condition. |
| title | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.19416 |