Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911110596132864 |
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| author | Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo |
| author_facet | Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo |
| contents | We generalize the one-dimensional population model of Anguige \& Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller \& \v Sverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition Choi, Hyung Jun Kim, Seonghak Koh, Youngwoo Analysis of PDEs 35M13, 35K59, 35D30, 92D25 We generalize the one-dimensional population model of Anguige \& Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller \& \v Sverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. |
| title | Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition |
| topic | Analysis of PDEs 35M13, 35K59, 35D30, 92D25 |
| url | https://arxiv.org/abs/2409.04689 |