Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912259313238016 |
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| author | Ahmedou, Mohameden Bartsch, Thomas Hu, Zhengni |
| author_facet | Ahmedou, Mohameden Bartsch, Thomas Hu, Zhengni |
| contents | We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -Δ_g u +βu =λ\left(\frac{Ve^u}{\int_Σ Ve^u d v_g}-1\right), &\text { in } \mathringΣ\\ \partial_{ ν_g} u=0, &\text { on } \partial Σ\end{array} \right.,\] on a compact Riemann surface $(Σ, g)$ of unit area, with interior $\mathringΣ$ and smooth boundary $\partial Σ$. Here, $Δ_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(Σ, g)$, and $ν_g$ the unit outward normal to $\partial Σ$ and $λ$ and $β$ are non-negative parameters, $V$ is non-negative with finite zero set.
For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $λ$ approaches the critical values $4πm$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces Ahmedou, Mohameden Bartsch, Thomas Hu, Zhengni Analysis of PDEs 35J57, 58J05 We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -Δ_g u +βu =λ\left(\frac{Ve^u}{\int_Σ Ve^u d v_g}-1\right), &\text { in } \mathringΣ\\ \partial_{ ν_g} u=0, &\text { on } \partial Σ\end{array} \right.,\] on a compact Riemann surface $(Σ, g)$ of unit area, with interior $\mathringΣ$ and smooth boundary $\partial Σ$. Here, $Δ_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(Σ, g)$, and $ν_g$ the unit outward normal to $\partial Σ$ and $λ$ and $β$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $λ$ approaches the critical values $4πm$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem. |
| title | Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces |
| topic | Analysis of PDEs 35J57, 58J05 |
| url | https://arxiv.org/abs/2409.00519 |