Nondegeneracy of blow-up points for the parabolic Keller–Segel system
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 851-875.

Dans cet article, nous étudions le système parabolique de Keller–Segel

{u t =·(u-u m v)dansΩ×(0,T),Γv t =Δv-λv+udansΩ×(0,T),
avec Ω un domaine de N , N1, où m,Γ>0, λ0 sont des constantes et T>0. Lorsque Ω 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

{u t =·(u-u m v)inΩ×(0,T),Γv t =Δv-λv+uinΩ×(0,T),
in a domain Ω of N with N1, where m,Γ>0, λ0 are constants and T>0. When Ω 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.

DOI : 10.1016/j.anihpc.2013.07.007
Classification : 35B44, 35K45, 92C17
Mots clés : Keller–Segel system, Blow-up, Nondegeneracy
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     title = {Nondegeneracy of blow-up points for the parabolic {Keller{\textendash}Segel} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {851--875},
     publisher = {Elsevier},
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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) no. 4, pp. 851-875. doi : 10.1016/j.anihpc.2013.07.007. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.007/

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