Well-Posedness for Quintic Energy Critical Wave in 3D Cylindrical Convex Domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914754552922112 |
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| author | Len, Meas |
| author_facet | Len, Meas |
| contents | In this paper, we establish the well-posedness in energy space for the quintic energy critical wave inside a cylindrical convex domain $Ω\subset\mathbb{R}^3$ with smooth boundary $\partialΩ\neq\emptyset$. The key tools to prove local well-posedness are the dispersive estimates obtained in \cite{L,L1,L3} and the Strichartz estimates in \cite{L2}. We point out that our result on the local and global existence of the solution to the wave equation in the cylindrical domain setting interpolates between that of in Euclidean space $\mathbb{R}^3$ (see \cite{MG}) and in any bounded domains in $\mathbb{R}^3$ (see \cite{BLP}). Moreover, the result of the Strichartz estimates in our setting is strong enough when combined with the arguments in \cite{BLP,SS2} so that we can extend local to global well-posedness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_09611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-Posedness for Quintic Energy Critical Wave in 3D Cylindrical Convex Domains Len, Meas Analysis of PDEs Primary 35L71, 35L05, Secondary 58J45, 42B25 In this paper, we establish the well-posedness in energy space for the quintic energy critical wave inside a cylindrical convex domain $Ω\subset\mathbb{R}^3$ with smooth boundary $\partialΩ\neq\emptyset$. The key tools to prove local well-posedness are the dispersive estimates obtained in \cite{L,L1,L3} and the Strichartz estimates in \cite{L2}. We point out that our result on the local and global existence of the solution to the wave equation in the cylindrical domain setting interpolates between that of in Euclidean space $\mathbb{R}^3$ (see \cite{MG}) and in any bounded domains in $\mathbb{R}^3$ (see \cite{BLP}). Moreover, the result of the Strichartz estimates in our setting is strong enough when combined with the arguments in \cite{BLP,SS2} so that we can extend local to global well-posedness. |
| title | Well-Posedness for Quintic Energy Critical Wave in 3D Cylindrical Convex Domains |
| topic | Analysis of PDEs Primary 35L71, 35L05, Secondary 58J45, 42B25 |
| url | https://arxiv.org/abs/2404.09611 |