Nondegeneracy of blow-up points for the parabolic Keller–Segel system
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, p. 851-875
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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.
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.
DOI : https://doi.org/10.1016/j.anihpc.2013.07.007
Classification:  35B44,  35K45,  92C17
Keywords: Keller–Segel system, Blow-up, Nondegeneracy
@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},
     publisher = {Elsevier},
     volume = {31},
     number = {4},
     year = {2014},
     pages = {851-875},
     doi = {10.1016/j.anihpc.2013.07.007},
     zbl = {1302.35075},
     mrnumber = {3249815},
     language = {en},
     url = {http://www.numdam.org/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, Volume 31 (2014) no. 4, pp. 851-875. doi : 10.1016/j.anihpc.2013.07.007. http://www.numdam.org/item/AIHPC_2014__31_4_851_0/

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