Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866909068888637440 |
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| author | Lafleche, Laurent Salem, Samir |
| author_facet | Lafleche, Laurent Salem, Samir |
| contents | This article studies the aggregation diffusion equation \[
\partial_tρ= Δ^\fracα{2} ρ+ λ\,\mathrm{div}((K*ρ)ρ), \] where $Δ^\fracα{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^β}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case $β< α$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $β= α$ for an $L^1_k\cap L\ln L$ initial condition with small mass. In the aggregation dominated case $β> α$, we prove global or local well-posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial data. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_06155 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up Lafleche, Laurent Salem, Samir Analysis of PDEs 35R11, 35A01, 35A02, 35B44, 35B40 This article studies the aggregation diffusion equation \[ \partial_tρ= Δ^\fracα{2} ρ+ λ\,\mathrm{div}((K*ρ)ρ), \] where $Δ^\fracα{2}$ denotes the fractional Laplacian and $K = \frac{x}{|x|^β}$ is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case $β< α$ we prove global well-posedness for an $L^1_k$ initial condition, and in the fair competition case $β= α$ for an $L^1_k\cap L\ln L$ initial condition with small mass. In the aggregation dominated case $β> α$, we prove global or local well-posedness for an $L^p$ initial condition, depending on some smallness condition on the $L^p$ norm of the initial data. We also prove that finite time blow-up of even solutions occurs under some initial mass concentration criteria. |
| title | Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up |
| topic | Analysis of PDEs 35R11, 35A01, 35A02, 35B44, 35B40 |
| url | https://arxiv.org/abs/1809.06155 |