Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915473426219008 |
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| author | Xu, Guixiang Yang, Pengxuan |
| author_facet | Xu, Guixiang Yang, Pengxuan |
| contents | We consider the defocusing, cubic nonlinear wave equation with zero Dirichlet boundary value in the exterior $Ω= \mathbb{R}^3\backslash \bar{ B}(0,1)$. We make use of the distorted Fourier transform in \cite{LiSZ:NLS, Taylor:PDE:II} to establish the dispersive estimate and the global-in-time (endpoint) Strichartz estimate of the linear wave equation outside of the ball with radial data. As an application, we combine the Fourier truncation method as those in \cite{Bourgain98:FTM, GallPlan03:NLW, KenigPV00:NLW} with the energy method to show global well-posedness of radial solution to the defoucusing, cubic nonlinear wave equation outside of a ball in the Sobolev space $\left(\dot H^{s}_{D}(Ω) \cap L^4(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ with $s>3/4$. To the best of the author's knowledge, it is first result about low regularity of semilinear wave equation with zero Dirichlet boundary value on the exterior domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data Xu, Guixiang Yang, Pengxuan Analysis of PDEs Primary: 35L05, Secondary: 35E05, 35L20 We consider the defocusing, cubic nonlinear wave equation with zero Dirichlet boundary value in the exterior $Ω= \mathbb{R}^3\backslash \bar{ B}(0,1)$. We make use of the distorted Fourier transform in \cite{LiSZ:NLS, Taylor:PDE:II} to establish the dispersive estimate and the global-in-time (endpoint) Strichartz estimate of the linear wave equation outside of the ball with radial data. As an application, we combine the Fourier truncation method as those in \cite{Bourgain98:FTM, GallPlan03:NLW, KenigPV00:NLW} with the energy method to show global well-posedness of radial solution to the defoucusing, cubic nonlinear wave equation outside of a ball in the Sobolev space $\left(\dot H^{s}_{D}(Ω) \cap L^4(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ with $s>3/4$. To the best of the author's knowledge, it is first result about low regularity of semilinear wave equation with zero Dirichlet boundary value on the exterior domain. |
| title | Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data |
| topic | Analysis of PDEs Primary: 35L05, Secondary: 35E05, 35L20 |
| url | https://arxiv.org/abs/2406.05614 |