Mass threshold for global existence in chemotaxis systems with critical flux limitation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908468650180608 |
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| author | Mao, Xuan Wang, Hengling Yan, Jianlu |
| author_facet | Mao, Xuan Wang, Hengling Yan, Jianlu |
| contents | This paper investigates the flux-limited chemotaxis system, proposed by Kohatsu and Senba~(2025), \begin{equation*}
\begin{cases} u_t = Δu -\nabla\cdot(u|\nabla v|^{α-2}\nabla v),\\ \:\:0=Δv + u,
\end{cases} \end{equation*} posed in the unit ball of $\mathbb{R}^N$ for some $N\geq2$, subject to no-flux and homogeneous Dirichlet boundary conditions. Due to precedents, e.g., Tello (2022) and Winkler (2022), the exponent $α= \frac{N}{N-1}$ is the threshold for finite-time blow-up under symmetry assumptions. We further find that under the framework of radially symmetric solutions, the system with critical flux limitation admits a globally bounded weak solution if and only if initial mass is strictly less than $ω_N \big(\frac{N^2}{N-1}\big)^{N-1}$, where $ω_N$ denotes the measure of the unit sphere $\mathbb{S}^{N-1}$. Asymptotic behaviors are also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mass threshold for global existence in chemotaxis systems with critical flux limitation Mao, Xuan Wang, Hengling Yan, Jianlu Analysis of PDEs This paper investigates the flux-limited chemotaxis system, proposed by Kohatsu and Senba~(2025), \begin{equation*} \begin{cases} u_t = Δu -\nabla\cdot(u|\nabla v|^{α-2}\nabla v),\\ \:\:0=Δv + u, \end{cases} \end{equation*} posed in the unit ball of $\mathbb{R}^N$ for some $N\geq2$, subject to no-flux and homogeneous Dirichlet boundary conditions. Due to precedents, e.g., Tello (2022) and Winkler (2022), the exponent $α= \frac{N}{N-1}$ is the threshold for finite-time blow-up under symmetry assumptions. We further find that under the framework of radially symmetric solutions, the system with critical flux limitation admits a globally bounded weak solution if and only if initial mass is strictly less than $ω_N \big(\frac{N^2}{N-1}\big)^{N-1}$, where $ω_N$ denotes the measure of the unit sphere $\mathbb{S}^{N-1}$. Asymptotic behaviors are also considered. |
| title | Mass threshold for global existence in chemotaxis systems with critical flux limitation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.19866 |