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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2507.15407 |
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| _version_ | 1866911067125317632 |
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| author | Ervedoza, Sylvain Tendani-Soler, Adrien |
| author_facet | Ervedoza, Sylvain Tendani-Soler, Adrien |
| contents | In this article, we provide a description of the reachable space for the heat equation with various lower order terms, set in the euclidean ball of $\mathbb{R}^d$ centered at $0$ and of radius one and controlled from the whole external boundary. Namely, we consider the case of linear heat equations with lower order terms of order $0$ and $1$, and the case of a semilinear heat equations. In the linear case, we prove that any function which can be extended as an holomorphic function in a set of the form $Ω_α= \{ z\in\mathbb{C}^d \big| |\Re(z)| + α|\Im(z)| < 1\}$ for some $α\in (0,1)$ and which admits a continuous extension up to $\overlineΩ_α$ belongs to the reachable space. In the semilinear case, we prove a similar result for sufficiently small data. Our proofs are based on well-posedness results for the heat equation in a suitable space of holomorphic functions over $Ω_α$ for $α> 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the reachable space for parabolic equations Ervedoza, Sylvain Tendani-Soler, Adrien Analysis of PDEs In this article, we provide a description of the reachable space for the heat equation with various lower order terms, set in the euclidean ball of $\mathbb{R}^d$ centered at $0$ and of radius one and controlled from the whole external boundary. Namely, we consider the case of linear heat equations with lower order terms of order $0$ and $1$, and the case of a semilinear heat equations. In the linear case, we prove that any function which can be extended as an holomorphic function in a set of the form $Ω_α= \{ z\in\mathbb{C}^d \big| |\Re(z)| + α|\Im(z)| < 1\}$ for some $α\in (0,1)$ and which admits a continuous extension up to $\overlineΩ_α$ belongs to the reachable space. In the semilinear case, we prove a similar result for sufficiently small data. Our proofs are based on well-posedness results for the heat equation in a suitable space of holomorphic functions over $Ω_α$ for $α> 1$. |
| title | On the reachable space for parabolic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.15407 |