Fast energy decay for 2-D wave equation with localized damping near spatial infinity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915498099212288 |
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| author | Ikehata, Ryo |
| author_facet | Ikehata, Ryo |
| contents | We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the results for the two-dimensional exterior mixed problem case, the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincaré-type inequalities in all spaces and the finite propagation property of the solution to construct an estimate formula. In the two-dimensional case, when comparing the problem in the whole space with that in the exterior domain, we find that there is a significant difference in the sense that the former requires a logarithmic correction to the energy decay rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast energy decay for 2-D wave equation with localized damping near spatial infinity Ikehata, Ryo Analysis of PDEs 35L05, 35B45, 35L15 We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the results for the two-dimensional exterior mixed problem case, the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincaré-type inequalities in all spaces and the finite propagation property of the solution to construct an estimate formula. In the two-dimensional case, when comparing the problem in the whole space with that in the exterior domain, we find that there is a significant difference in the sense that the former requires a logarithmic correction to the energy decay rate. |
| title | Fast energy decay for 2-D wave equation with localized damping near spatial infinity |
| topic | Analysis of PDEs 35L05, 35B45, 35L15 |
| url | https://arxiv.org/abs/2509.13645 |