Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917913549602816 |
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| author | Kohatsu, Shohei |
| author_facet | Kohatsu, Shohei |
| contents | The flux-limited Keller--Segel system \begin{align*}
\begin{cases}
u_t = Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v),
\\[]
v_t = Δv - v + u^θ
\end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$. In the case that $θ\le 1$, existence of global bounded weak solutions was established in the previous work (arXiv:2501.04370 ; to be appear in Proceedings of the conference "Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities."). The purpose of this paper is to prove that global bounded weak solutions can also be constructed in the case $θ> 1$ with a smallness condition on $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production Kohatsu, Shohei Analysis of PDEs Primary: 35B45, Secondary: 35Q92, 35D30, 92C17 The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = Δv - v + u^θ \end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$. In the case that $θ\le 1$, existence of global bounded weak solutions was established in the previous work (arXiv:2501.04370 ; to be appear in Proceedings of the conference "Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities."). The purpose of this paper is to prove that global bounded weak solutions can also be constructed in the case $θ> 1$ with a smallness condition on $p$. |
| title | Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production |
| topic | Analysis of PDEs Primary: 35B45, Secondary: 35Q92, 35D30, 92C17 |
| url | https://arxiv.org/abs/2502.02882 |