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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2212.01029 |
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| _version_ | 1866911248730292224 |
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| author | Inami, Kotaro Suzuki, Soichiro |
| author_facet | Inami, Kotaro Suzuki, Soichiro |
| contents | We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d = 1$, Green (2020) established that the exponential decay for $s \geq 2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s \geq 2$. In addition, we also prove that the exponential decay holds for $0 < s < 2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01029 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients Inami, Kotaro Suzuki, Soichiro Analysis of PDEs 350L, 42A38 We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d = 1$, Green (2020) established that the exponential decay for $s \geq 2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s \geq 2$. In addition, we also prove that the exponential decay holds for $0 < s < 2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition. |
| title | Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients |
| topic | Analysis of PDEs 350L, 42A38 |
| url | https://arxiv.org/abs/2212.01029 |