On a fractional semilinear Neumann problem arising in Chemotaxis
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911163104624640 |
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| author | Cinti, Eleonora Talluri, Matteo |
| author_facet | Cinti, Eleonora Talluri, Matteo |
| contents | We study a semilinear and nonlocal Neumann problem, which is the fractional analogue of the problem considered by Lin--Ni--Takagi in the '80s. The model under consideration arises in the description of stationary configurations of the Keller--Segel model for chemotaxis, when a nonlocal diffusion for the concentration of the chemical is considered. In particular, we extend to any fractional power $s\in (0,1)$ of the Laplacian (with homogeneous Neumann boundary conditions) the results obtained in [20] for $s=1/2$. We prove existence and some qualitative properties of non--constant solutions when the diffusion parameter $\varepsilon$ is small enough, and on the other hand, we show that for $\varepsilon$ large enough any solution must be necessarily constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12181 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a fractional semilinear Neumann problem arising in Chemotaxis Cinti, Eleonora Talluri, Matteo Analysis of PDEs 35R11, 35B09, 35B45, 35A15, 60G22 We study a semilinear and nonlocal Neumann problem, which is the fractional analogue of the problem considered by Lin--Ni--Takagi in the '80s. The model under consideration arises in the description of stationary configurations of the Keller--Segel model for chemotaxis, when a nonlocal diffusion for the concentration of the chemical is considered. In particular, we extend to any fractional power $s\in (0,1)$ of the Laplacian (with homogeneous Neumann boundary conditions) the results obtained in [20] for $s=1/2$. We prove existence and some qualitative properties of non--constant solutions when the diffusion parameter $\varepsilon$ is small enough, and on the other hand, we show that for $\varepsilon$ large enough any solution must be necessarily constant. |
| title | On a fractional semilinear Neumann problem arising in Chemotaxis |
| topic | Analysis of PDEs 35R11, 35B09, 35B45, 35A15, 60G22 |
| url | https://arxiv.org/abs/2507.12181 |