On the system of $2$-D elastic waves with critical space dependent damping
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915498586800128 |
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| author | Charão, Ruy Coimbra Ikehata, Ryo |
| author_facet | Charão, Ruy Coimbra Ikehata, Ryo |
| contents | We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as time goes to infinity. In the $2$-D case we do not have any suitable Hardy type inequality, so generally one has no idea to establish optimal energy decay. We develope a special type of multiplier method combined with some estimates brought by the $2$-D Newton potential belonging to the usual Laplacian $-Δ$, not the operator $-a^2Δ- (b^{2}-a^{2})\nabla {\rm div}$ itself. The property of finite speed propagation is important to get results for this system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the system of $2$-D elastic waves with critical space dependent damping Charão, Ruy Coimbra Ikehata, Ryo Analysis of PDEs Functional Analysis Primary 35L52, 35L05, Secondary 35B40, 35B45, 35J05 We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as time goes to infinity. In the $2$-D case we do not have any suitable Hardy type inequality, so generally one has no idea to establish optimal energy decay. We develope a special type of multiplier method combined with some estimates brought by the $2$-D Newton potential belonging to the usual Laplacian $-Δ$, not the operator $-a^2Δ- (b^{2}-a^{2})\nabla {\rm div}$ itself. The property of finite speed propagation is important to get results for this system. |
| title | On the system of $2$-D elastic waves with critical space dependent damping |
| topic | Analysis of PDEs Functional Analysis Primary 35L52, 35L05, Secondary 35B40, 35B45, 35J05 |
| url | https://arxiv.org/abs/2503.06854 |