Nondegeneracy of blow-up points for the parabolic Keller–Segel system

Mizoguchi, Noriko; Souplet, Philippe

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 851-875 / Harvested from Numdam

Résumé
Abstract

Dans cet article, nous étudions le système parabolique de Keller–Segel ${\begin{matrix} u_{t} = \nabla \cdot (\nabla u - u^{m} \nabla v) & dans Ω \times (0, T), \\ Γ v_{t} = Δ v - λ v + u & dans Ω \times (0, T), \end{matrix}$ avec Ω un domaine de $ℝ^{N}$ , $N ⩾ 1$ , où $m, Γ > 0$ , $λ ⩾ 0$ sont des constantes et $T > 0$ . Lorsque $Ω \neq ℝ^{N}$ , les conditions aux limites de Neumann sont prescrites sur le bord. Sous des hypothèses convenables, nous prouvons la non-dégénérescence locale des points d'explosion. Ce résultat semble nouveau même dans le cas du système de Keller–Segel classique ( $m = 1$ ). Des estimations inférieures globales de la vitesse d'explosion sont également obtenues. Dans le cas singulier $0 < m < 1$ , nous établissons les propriétés nécessaires d'existence locale et de régularité.

This paper is concerned with the parabolic Keller–Segel system ${\begin{matrix} u_{t} = \nabla \cdot (\nabla u - u^{m} \nabla v) & in Ω \times (0, T), \\ Γ v_{t} = Δ v - λ v + u & in Ω \times (0, T), \end{matrix}$ in a domain Ω of $ℝ^{N}$ with $N ⩾ 1$ , where $m, Γ > 0$ , $λ ⩾ 0$ are constants and $T > 0$ . When $Ω \neq ℝ^{N}$ , we impose the Neumann boundary conditions on the boundary. Under suitable assumptions, we prove the local nondegeneracy of blow-up points. This seems new even for the classical Keller–Segel system ( $m = 1$ ). Lower global blow-up estimates are also obtained. In the singular case $0 < m < 1$ , as a prerequisite, local existence and regularity properties are established.

Publié le : 2014-01-01

MR 3249815 | Zbl 1302.35075

DOI : https://doi.org/10.1016/j.anihpc.2013.07.007
Classification: 35B44, 35K45, 92C17

@article{AIHPC_2014__31_4_851_0,
     author = {Mizoguchi, Noriko and Souplet, Philippe},
     title = {Nondegeneracy of blow-up points for the parabolic Keller--Segel system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {851-875},
     doi = {10.1016/j.anihpc.2013.07.007},
     mrnumber = {3249815},
     zbl = {1302.35075},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_4_851_0}
}

Mizoguchi, Noriko; Souplet, Philippe. Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 851-875. doi : 10.1016/j.anihpc.2013.07.007. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_4_851_0/

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