Global boundedness for a two-dimensional doubly degenerate nutrient taxis system
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917827336732672 |
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| author | Zhang, Zhiguang Li, Yuxiang |
| author_facet | Zhang, Zhiguang Li, Yuxiang |
| contents | This paper is concerned with the doubly degenerate nutrient taxis system $u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ uv$ and $v_t=Δv-u v$ for some $l \geqslant 1$, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset \mathbb{R}^n$ $(n \leqslant 2)$. Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if $l \in[1,3]$, then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for $l \in[1,\infty)$. Notably, the solution remains uniformly bounded when $l \in[1,\infty)$ in one dimension or $l \in(1,3]$ in two dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global boundedness for a two-dimensional doubly degenerate nutrient taxis system Zhang, Zhiguang Li, Yuxiang Analysis of PDEs This paper is concerned with the doubly degenerate nutrient taxis system $u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ uv$ and $v_t=Δv-u v$ for some $l \geqslant 1$, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset \mathbb{R}^n$ $(n \leqslant 2)$. Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if $l \in[1,3]$, then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for $l \in[1,\infty)$. Notably, the solution remains uniformly bounded when $l \in[1,\infty)$ in one dimension or $l \in(1,3]$ in two dimensions. |
| title | Global boundedness for a two-dimensional doubly degenerate nutrient taxis system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.01133 |