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Article Dans Une Revue Communications in Mathematical Sciences Année : 2019

Fractional Keller–Segel Equation: Global Well-posedness and Finite Time Blow-up

Équation de Keller–Segel fractionnaire : Existence et unicité globales et explosion en temps fini

Résumé

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.
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Dates et versions

hal-01875506 , version 1 (17-09-2018)

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Laurent Lafleche, Samir Salem. Fractional Keller–Segel Equation: Global Well-posedness and Finite Time Blow-up. Communications in Mathematical Sciences, 2019, 17 (8), pp.2055-2087. ⟨10.4310/CMS.2019.v17.n8.a1⟩. ⟨hal-01875506⟩
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