A critical nonlinearity for blow-up in a higher-dimensional chemotaxis system with indirect signal production
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915053495648256 |
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| author | Zhao, Yiheng |
| author_facet | Zhao, Yiheng |
| contents | The Neumann problem in balls $Ω\subset\mathbb{R}^n$, $n\in\{3,4\}$, for the chemotaxis system \begin{equation*} \left\{ \begin{array}{ll} u_t = Δu - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - μ^{(w)}(t) + w, \quad μ^{(w)}(t) = \frac{1}{|Ω|}\int_Ωw \\[1mm] w_t = Δw - w + f(u), \end{array} \right. \end{equation*} is considered. Under the assumption that $f\in C^1([0,\infty))$ is such that $f(ξ) \ge kξ^σ$ for all $ξ\ge 0$ and some $k>0$ and $σ>\frac{4}{n}$, it is shown that finite-time blow-up occurs for some radially symmetric solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A critical nonlinearity for blow-up in a higher-dimensional chemotaxis system with indirect signal production Zhao, Yiheng Analysis of PDEs 35B44 (primary), 35B33, 35K57, 35Q92, 92C17 (secondary) The Neumann problem in balls $Ω\subset\mathbb{R}^n$, $n\in\{3,4\}$, for the chemotaxis system \begin{equation*} \left\{ \begin{array}{ll} u_t = Δu - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - μ^{(w)}(t) + w, \quad μ^{(w)}(t) = \frac{1}{|Ω|}\int_Ωw \\[1mm] w_t = Δw - w + f(u), \end{array} \right. \end{equation*} is considered. Under the assumption that $f\in C^1([0,\infty))$ is such that $f(ξ) \ge kξ^σ$ for all $ξ\ge 0$ and some $k>0$ and $σ>\frac{4}{n}$, it is shown that finite-time blow-up occurs for some radially symmetric solutions. |
| title | A critical nonlinearity for blow-up in a higher-dimensional chemotaxis system with indirect signal production |
| topic | Analysis of PDEs 35B44 (primary), 35B33, 35K57, 35Q92, 92C17 (secondary) |
| url | https://arxiv.org/abs/2412.05909 |