Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910115772235776 |
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| author | Shibata, Osuke Yokota, Tomomi |
| author_facet | Shibata, Osuke Yokota, Tomomi |
| contents | In a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*}
u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v),
\quad
v_t = Δv + g(u,v),
\quad x\in Ω, \ t>0, \end{align*} is considered under the assumptions that $D(1,s)=0$ and that $h(0,s)=h(1,s)=0$. Here, initial data $u_0$ and $v_0$ have suitable regularity and satisfy $0\le u_0\le 1$ and $v_0\ge 0$ with $\nabla v_0 \cdot ν|_{\partial Ω} = 0$. It is proved that there exists a global weak solution such that $0\le u\le 1$ and $v\ge 0$. Moreover, when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$ and additional conditions on $D$, $h$ and $g$ are assumed, uniqueness of global weak solutions with the mass conservation law $\int_Ωu(x,t) \, dx = \int_Ωu_0(x) \, dx$ is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density Shibata, Osuke Yokota, Tomomi Analysis of PDEs 35K65, 35K55, 34B15, 34C25 In a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v), \quad v_t = Δv + g(u,v), \quad x\in Ω, \ t>0, \end{align*} is considered under the assumptions that $D(1,s)=0$ and that $h(0,s)=h(1,s)=0$. Here, initial data $u_0$ and $v_0$ have suitable regularity and satisfy $0\le u_0\le 1$ and $v_0\ge 0$ with $\nabla v_0 \cdot ν|_{\partial Ω} = 0$. It is proved that there exists a global weak solution such that $0\le u\le 1$ and $v\ge 0$. Moreover, when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$ and additional conditions on $D$, $h$ and $g$ are assumed, uniqueness of global weak solutions with the mass conservation law $\int_Ωu(x,t) \, dx = \int_Ωu_0(x) \, dx$ is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting |
| title | Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density |
| topic | Analysis of PDEs 35K65, 35K55, 34B15, 34C25 |
| url | https://arxiv.org/abs/2604.07978 |