Determining a parabolic system by boundary observation of its non-negative solutions with biological applications
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917645971881984 |
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| author | Liu, Hongyu Lo, Catharine W. K. |
| author_facet | Liu, Hongyu Lo, Catharine W. K. |
| contents | In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in PDE analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13045 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Determining a parabolic system by boundary observation of its non-negative solutions with biological applications Liu, Hongyu Lo, Catharine W. K. Analysis of PDEs 35R30, 35B09, 35K51 In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in PDE analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities. |
| title | Determining a parabolic system by boundary observation of its non-negative solutions with biological applications |
| topic | Analysis of PDEs 35R30, 35B09, 35K51 |
| url | https://arxiv.org/abs/2303.13045 |