Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.19318 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917270487302144 |
|---|---|
| author | Le, Minh |
| author_facet | Le, Minh |
| contents | This paper deals with the problem of global solvability and boundedness of classical solutions to a fully parabolic chemotaxis system with singular sensitivity in any dimensional setting. In particular, We show that the system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \dfrac{u}{v} \nabla v \right), \\ v_t = Δv - v + u, \end{cases} \end{equation*} posed in a bounded domain $Ω\subset \mathbb{R}^n$ with $n \geq 3$, admits a global bounded classical solution provided that $χ\in (0,χ_0)$ with $χ_0 > \sqrt{\frac{2}{n}}$ can be determined explicitly. This result extends several existing works, which established global boundedness under the more restrictive condition $χ< \sqrt{\frac{2}{n}}$, and shows that this threshold is not an optimal upper bound for preventing blow-up. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19318 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improvement toward global boundedness in a fully parabolic chemotaxis with singular sensitivity in any dimension Le, Minh Analysis of PDEs This paper deals with the problem of global solvability and boundedness of classical solutions to a fully parabolic chemotaxis system with singular sensitivity in any dimensional setting. In particular, We show that the system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot \left( \dfrac{u}{v} \nabla v \right), \\ v_t = Δv - v + u, \end{cases} \end{equation*} posed in a bounded domain $Ω\subset \mathbb{R}^n$ with $n \geq 3$, admits a global bounded classical solution provided that $χ\in (0,χ_0)$ with $χ_0 > \sqrt{\frac{2}{n}}$ can be determined explicitly. This result extends several existing works, which established global boundedness under the more restrictive condition $χ< \sqrt{\frac{2}{n}}$, and shows that this threshold is not an optimal upper bound for preventing blow-up. |
| title | An improvement toward global boundedness in a fully parabolic chemotaxis with singular sensitivity in any dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.19318 |