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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.28350 |
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| _version_ | 1866908921037324288 |
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| author | Luo, Xinyue Yamamoto, Masahiro Cheng, Jin |
| author_facet | Luo, Xinyue Yamamoto, Masahiro Cheng, Jin |
| contents | We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain $Ω$ over a time interval $(0,T)$, which governs the water density $u(x,t)$ and the vegetation biomass density $v(x,t)$ for $x\inΩ$ and $0<t<T$. In this system, called the Klausmeier-Gray-Scott model, we assume that an unknown source depends only on the spatial variable and appears in the reaction-diffusion equation for $u$.
The main subject is the inverse source problem of determining a source term from limited data on $(u,v)$. We establish two kinds of stability estimates by means of Carleman estimates. First, a Carleman estimate with a singular weight yields a Lipschitz stability estimate for the inverse source problem from data consisting of a snapshot $u(\cdot,t_0)$ in $Ω$ and $(u,v)$ in a subdomain $ω$ over a time interval. Second, without assuming boundary data, we prove a Hölder stability estimate in any interior subdomain $Ω_0$ satisfying $\overline{Ω_0}\subsetΩ$. We further study how much the observation data can be reduced while preserving uniqueness and stability in the inverse problem under suitable additional conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse source problems with reduced interior data for a coupled reaction-diffusion system Luo, Xinyue Yamamoto, Masahiro Cheng, Jin Analysis of PDEs We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain $Ω$ over a time interval $(0,T)$, which governs the water density $u(x,t)$ and the vegetation biomass density $v(x,t)$ for $x\inΩ$ and $0<t<T$. In this system, called the Klausmeier-Gray-Scott model, we assume that an unknown source depends only on the spatial variable and appears in the reaction-diffusion equation for $u$. The main subject is the inverse source problem of determining a source term from limited data on $(u,v)$. We establish two kinds of stability estimates by means of Carleman estimates. First, a Carleman estimate with a singular weight yields a Lipschitz stability estimate for the inverse source problem from data consisting of a snapshot $u(\cdot,t_0)$ in $Ω$ and $(u,v)$ in a subdomain $ω$ over a time interval. Second, without assuming boundary data, we prove a Hölder stability estimate in any interior subdomain $Ω_0$ satisfying $\overline{Ω_0}\subsetΩ$. We further study how much the observation data can be reduced while preserving uniqueness and stability in the inverse problem under suitable additional conditions. |
| title | Inverse source problems with reduced interior data for a coupled reaction-diffusion system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.28350 |