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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.10586 |
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| _version_ | 1866910789239046144 |
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| author | Menacho, Joaquín Pellicer, Marta Solà-Morales, J. |
| author_facet | Menacho, Joaquín Pellicer, Marta Solà-Morales, J. |
| contents | In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can also be seen as a weakly damped wave equation with certain boundary conditions. We are interested in the long-time behaviour of its solutions. To be precise, we will prove that the decay of the solutions to this problem is of exponential form, where the optimal decay rate exponent is given by the dominant eigenvalue of the corresponding operator. This eigenvalue can be obtained as a particular solution of a system of transcendental equations. A complete description of the spectrum of the operator is provided, together with a comprehensive analysis of the corresponding eigenfunctions and their geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time behaviour of the correlated random walk system Menacho, Joaquín Pellicer, Marta Solà-Morales, J. Analysis of PDEs 35L40, 35P15, 35Q92 In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can also be seen as a weakly damped wave equation with certain boundary conditions. We are interested in the long-time behaviour of its solutions. To be precise, we will prove that the decay of the solutions to this problem is of exponential form, where the optimal decay rate exponent is given by the dominant eigenvalue of the corresponding operator. This eigenvalue can be obtained as a particular solution of a system of transcendental equations. A complete description of the spectrum of the operator is provided, together with a comprehensive analysis of the corresponding eigenfunctions and their geometry. |
| title | Long-time behaviour of the correlated random walk system |
| topic | Analysis of PDEs 35L40, 35P15, 35Q92 |
| url | https://arxiv.org/abs/2501.10586 |