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

Abstract

This paper is concerned with the parabolic Keller-Segel systemin a domain Ω of R N with N 1, where m, Γ > 0, λ 0 are constants and T > 0. When Ω = 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. © 2013 RésuméDans cet article, nous étudions le système parabolique de Keller-Segelles 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.

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