
Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up
Lafleche, Laurent; Salem, Samir (2018), Fractional Keller-Segel Equation: Global Well-posedness and Finite Time Blow-up. https://basepub.dauphine.fr/handle/123456789/18472
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Type
Document de travail / Working paperExternal document link
https://hal.archives-ouvertes.fr/hal-01875506Date
2018Series title
Cahier de recherche CEREMADE, Université Paris-DauphinePages
30
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Show full item recordAuthor(s)
Lafleche, Laurent
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Salem, Samir
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
This article studies the aggregation diffusion equation ∂ρ/∂t = ∆^(α/2) ρ + λ div((K * ρ)ρ), where ∆^(α/2) denotes the fractional Laplacian and K = x/|x|^a 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 a < α we prove global well-posedness for an L^1_k initial condition, and in the fair competition case a = α for an L^1_k ∩ L ln L initial condition. In the aggregation dominated case a > 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 condition. We also prove that finite time blow-up of even solutions occurs, under some initial mass concentration criteria.Subjects / Keywords
Keller-Segel Equation; Analysis of PDEsRelated items
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