Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913717895036928 |
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| author | Le, Minh |
| author_facet | Le, Minh |
| contents | It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(Ω\subset \mathbb{R}^2\): \begin{equation*}
\begin{cases}
u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, \qquad &\text{in } Ω\times (0,T_{\rm max}),
v_t = Δv - αv + βu, \qquad &\text{in } Ω\times (0,T_{\rm max}),
\end{cases} \end{equation*} where \( k \in (0,1) \), and \(χ, r, μ, α, β\) are positive parameters. Known results have already established the same conclusion for the parabolic-elliptic case. Here, we complement these findings by extending the result to the fully parabolic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two Le, Minh Analysis of PDEs It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(Ω\subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \frac{u}{v^k} \nabla v \right) + ru - μu^2, \qquad &\text{in } Ω\times (0,T_{\rm max}), v_t = Δv - αv + βu, \qquad &\text{in } Ω\times (0,T_{\rm max}), \end{cases} \end{equation*} where \( k \in (0,1) \), and \(χ, r, μ, α, β\) are positive parameters. Known results have already established the same conclusion for the parabolic-elliptic case. Here, we complement these findings by extending the result to the fully parabolic case. |
| title | Absence of blow-up in a fully parabolic chemotaxis system with weak singular sensitivity and logistic damping in dimension two |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.02346 |