A dimension-independent critical exponent in a nutrient taxis system
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911362376007680 |
|---|---|
| author | Winkler, Michael |
| author_facet | Winkler, Michael |
| contents | In a ball $Ω\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[
\left\{ \begin{array}{l}
u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm]
0 = Δv - uv,
\end{array} \right. \] is considered under no-flux boundary conditions for $u$, and for prescribed constant positive boundary data for $v$. Under the assumption that $D\in C^3([0,\infty))$ satisfies \[
D(ξ)\ge k_D (ξ+1)^{-α}
\qquad \mbox{for all } ξ\ge 0
\qquad \qquad
(\star) \] with some $α<1$ and some $k_D>0$, it is shown that for each nonnegative and radially symmetric $u_0\in \bigcup_{q>\max\{2,n\}} W^{1,q}(Ω)$, a uniquely determined global bounded classical solution exists.
This complements a previous result according to which given any positive $D\in C^3([0,\infty))$ fulfilling $D(ξ) \le K_D (ξ+1)^{-α}$ with some $α>1$ and $K_D>0$, one can find nonnegative radial initial data $u_0\in C_0^\infty(Ω)$ such that no global solution exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05338 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A dimension-independent critical exponent in a nutrient taxis system Winkler, Michael Analysis of PDEs 35B45, 35K51, 35Q92, 92C17 In a ball $Ω\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - uv, \end{array} \right. \] is considered under no-flux boundary conditions for $u$, and for prescribed constant positive boundary data for $v$. Under the assumption that $D\in C^3([0,\infty))$ satisfies \[ D(ξ)\ge k_D (ξ+1)^{-α} \qquad \mbox{for all } ξ\ge 0 \qquad \qquad (\star) \] with some $α<1$ and some $k_D>0$, it is shown that for each nonnegative and radially symmetric $u_0\in \bigcup_{q>\max\{2,n\}} W^{1,q}(Ω)$, a uniquely determined global bounded classical solution exists. This complements a previous result according to which given any positive $D\in C^3([0,\infty))$ fulfilling $D(ξ) \le K_D (ξ+1)^{-α}$ with some $α>1$ and $K_D>0$, one can find nonnegative radial initial data $u_0\in C_0^\infty(Ω)$ such that no global solution exists. |
| title | A dimension-independent critical exponent in a nutrient taxis system |
| topic | Analysis of PDEs 35B45, 35K51, 35Q92, 92C17 |
| url | https://arxiv.org/abs/2601.05338 |