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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04945 |
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| _version_ | 1866917636474929152 |
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| author | Duan, Yueliang Zhang, Can |
| author_facet | Duan, Yueliang Zhang, Can |
| contents | Given an equidistributed set in the whole Euclidean space, we have established in [1] that there exists a constant positive $C$ such that the observability inequality of diffusion equations holds for all $T\in]0,1[$, with an observability cost being of the form $Ce^{C/T}$. In this paper, for any small constant $\varepsilon>0$, we prove that there exists a nontrivial equidistributed set (in the sense that whose complementary set is unbounded), so that the above observability cost can be improved to a fast form of $Ce^{\varepsilon/T}$ for certain constant $C>0$. The proof is based on the strategy used in [1], as well as an interpolation inequality for gradients of solutions to elliptic equations obtained recently in [2]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04945 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Fast Observability for Diffusion Equations in $\mathbb R^N$ Duan, Yueliang Zhang, Can Analysis of PDEs Optimization and Control Given an equidistributed set in the whole Euclidean space, we have established in [1] that there exists a constant positive $C$ such that the observability inequality of diffusion equations holds for all $T\in]0,1[$, with an observability cost being of the form $Ce^{C/T}$. In this paper, for any small constant $\varepsilon>0$, we prove that there exists a nontrivial equidistributed set (in the sense that whose complementary set is unbounded), so that the above observability cost can be improved to a fast form of $Ce^{\varepsilon/T}$ for certain constant $C>0$. The proof is based on the strategy used in [1], as well as an interpolation inequality for gradients of solutions to elliptic equations obtained recently in [2]. |
| title | A Fast Observability for Diffusion Equations in $\mathbb R^N$ |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2404.04945 |