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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.05188 |
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| _version_ | 1866912396883263488 |
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| author | Xu, Guixiang Yang, Pengxuan You, Zhuohui |
| author_facet | Xu, Guixiang Yang, Pengxuan You, Zhuohui |
| contents | We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $Ω= \mathbb{R}^3\backslash \bar{B}(0,1)$. We combine the distorted Fourier truncation method in \cite{Bourgain98:FTM}, the global-in-time (endpoint) Strichartz estimates in \cite{XuYang:NLW} with the energy method in \cite{GallPlan03:NLW} to prove the global well-posedness of the radial solution to the defocusing, energy-subcriticial nonlinear wave equation outside of a ball in $\left(\dot H^{s}_{D}(Ω) \cap L^{p+1}(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ with $1-\frac{(p+3)(1-s_c)}{4(2p-3)}<s<1$, $s_c=\frac{3}{2}-\frac{2}{p-1} $, which extends the result for the cubic nonlinearity in \cite{XuYang:NLW} to the case $3<p<5$. Except from the argument in \cite{XuYang:NLW}, another new ingredient is that we need make use of the radial Sobolev inequality to deal with the super-conformal nonlinearity in addition to the Sobolev inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of the defocusing nonlinear wave equation outside of a ball with radial data for $3<p<5$ Xu, Guixiang Yang, Pengxuan You, Zhuohui Analysis of PDEs We continue the study of the Dirichlet boundary value problem of nonlinear wave equation with radial data in the exterior $Ω= \mathbb{R}^3\backslash \bar{B}(0,1)$. We combine the distorted Fourier truncation method in \cite{Bourgain98:FTM}, the global-in-time (endpoint) Strichartz estimates in \cite{XuYang:NLW} with the energy method in \cite{GallPlan03:NLW} to prove the global well-posedness of the radial solution to the defocusing, energy-subcriticial nonlinear wave equation outside of a ball in $\left(\dot H^{s}_{D}(Ω) \cap L^{p+1}(Ω) \right)\times \dot H^{s-1}_{D}(Ω)$ with $1-\frac{(p+3)(1-s_c)}{4(2p-3)}<s<1$, $s_c=\frac{3}{2}-\frac{2}{p-1} $, which extends the result for the cubic nonlinearity in \cite{XuYang:NLW} to the case $3<p<5$. Except from the argument in \cite{XuYang:NLW}, another new ingredient is that we need make use of the radial Sobolev inequality to deal with the super-conformal nonlinearity in addition to the Sobolev inequality. |
| title | Global well-posedness of the defocusing nonlinear wave equation outside of a ball with radial data for $3<p<5$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.05188 |