Well-posedness of Keller-Segel systems on compact metric graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916187671101440 |
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| author | Shemtaga, Hewan Shen, Wenxian Sukhtaiev, Selim |
| author_facet | Shemtaga, Hewan Shen, Wenxian Sukhtaiev, Selim |
| contents | Chemotaxis phenomena govern the directed movement of micro-organisms in response to chemical stimuli. In this paper, we investigate two Keller--Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic type Keller--Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19747 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness of Keller-Segel systems on compact metric graphs Shemtaga, Hewan Shen, Wenxian Sukhtaiev, Selim Analysis of PDEs Spectral Theory 35Q92, 92C17, 35P05 Chemotaxis phenomena govern the directed movement of micro-organisms in response to chemical stimuli. In this paper, we investigate two Keller--Segel systems of reaction-advection-diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic type Keller--Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions. |
| title | Well-posedness of Keller-Segel systems on compact metric graphs |
| topic | Analysis of PDEs Spectral Theory 35Q92, 92C17, 35P05 |
| url | https://arxiv.org/abs/2403.19747 |