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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.19833 |
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| _version_ | 1866910670456356864 |
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| author | Zhang, Zhiguang Li, Yuxiang |
| author_facet | Zhang, Zhiguang Li, Yuxiang |
| contents | In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=Δv-u v \end{cases} \end{align} in a smooth bounded domain $Ω\subset \mathbb{R}^2$, where $l \geqslant 1$. It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when $l = 2$, our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source Zhang, Zhiguang Li, Yuxiang Analysis of PDEs In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\ v_t=Δv-u v \end{cases} \end{align} in a smooth bounded domain $Ω\subset \mathbb{R}^2$, where $l \geqslant 1$. It is proved that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem \eqref{eq 0.1} possesses a global weak solution which is continuous in its first and essentially smooth in its second component. We point out that when $l = 2$, our result is consistent with that of [G. Li and M. Winkler, Analysis and Applications, (2024)]. |
| title | Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.19833 |