A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910692967186432 |
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| author | Tao, Youshan Winkler, Michael |
| author_facet | Tao, Youshan Winkler, Michael |
| contents | In bounded $n$-dimensional domains with $n\ge 3$, this manuscript considers an initial-boundary problem for a quasilinear chemotaxis system with indirect attractant production, as arising, inter alia, in the modeling of effects due to phenotypical heterogeneity in microbial populations. Under the assumption that the rates $D$ and $S$ of diffusion and cross-diffusion are suitably regular functions of the population density, essentially exhibiting asymptotic behavior of the form \[
D(ξ) \simeq ξ^{m-1}
\quad \mbox{and} \quad
S(ξ) \simeq ξ^σ,
\qquad ξ\simeq \infty, \] the identity \[
σ=m-1+\frac{4}{n}
\qquad \qquad (n\ge 3), \] is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation \[
σ=m+\frac{2}{n}
\qquad \qquad (n\le 2) \] is known to play a correspondingly pivotal role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production Tao, Youshan Winkler, Michael Analysis of PDEs 35B44 (primary), 35K57, 35K59, 35Q92, 92C17 (secondary) In bounded $n$-dimensional domains with $n\ge 3$, this manuscript considers an initial-boundary problem for a quasilinear chemotaxis system with indirect attractant production, as arising, inter alia, in the modeling of effects due to phenotypical heterogeneity in microbial populations. Under the assumption that the rates $D$ and $S$ of diffusion and cross-diffusion are suitably regular functions of the population density, essentially exhibiting asymptotic behavior of the form \[ D(ξ) \simeq ξ^{m-1} \quad \mbox{and} \quad S(ξ) \simeq ξ^σ, \qquad ξ\simeq \infty, \] the identity \[ σ=m-1+\frac{4}{n} \qquad \qquad (n\ge 3), \] is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation \[ σ=m+\frac{2}{n} \qquad \qquad (n\le 2) \] is known to play a correspondingly pivotal role. |
| title | A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production |
| topic | Analysis of PDEs 35B44 (primary), 35K57, 35K59, 35Q92, 92C17 (secondary) |
| url | https://arxiv.org/abs/2411.06475 |