Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition
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 i...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100728/type/journal_article |
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| _version_ | 1850089245920722944 |
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| author | Dongxiao Yu |
| author_facet | Dongxiao Yu |
| author_sort | Dongxiao Yu |
| collection | DOAJ |
| description | 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 [75]. 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 | Article |
| id | doaj-art-09e8a8b9c92d477a848b8322bfbd6641 |
| institution | DOAJ |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-09e8a8b9c92d477a848b8322bfbd66412025-08-20T02:42:49ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10072Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null conditionDongxiao Yu0https://orcid.org/0000-0001-9947-1942Department of Mathematics, https://ror.org/02vm5rt34Vanderbilt UniversityWe 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 [75]. 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.https://www.cambridge.org/core/product/identifier/S2050509425100728/type/journal_article35L7235B40 |
| spellingShingle | Dongxiao Yu Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition Forum of Mathematics, Sigma 35L72 35B40 |
| title | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| title_full | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| title_fullStr | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| title_full_unstemmed | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| title_short | Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| title_sort | timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition |
| topic | 35L72 35B40 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100728/type/journal_article |
| work_keys_str_mv | AT dongxiaoyu timelikeasymptoticsforglobalsolutionstoascalarquasilinearwaveequationsatisfyingtheweaknullcondition |