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) no. 4, pp. 851-875.

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

$$\{\begin{array}{cc}{u}_t=\nabla ·(\nabla u-u^m\nabla v)\hfill & \text{dans}\Omega \times (0,T),\hfill \\ \Gamma v_t=\Delta v-\lambda v+u\hfill & \text{dans}\Omega \times (0,T),\hfill \end{array}$$

avec Ω un domaine de ${ℝ}^N$, $N⩾1$, où $m,\Gamma >0$, $\lambda ⩾0$ sont des constantes et $T>0$. Lorsque $\Omega \ne {ℝ}^N$, 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

$$\{\begin{array}{cc}{u}_t=\nabla ·(\nabla u-u^m\nabla v)\hfill & \text{in}\Omega \times (0,T),\hfill \\ \Gamma v_t=\Delta v-\lambda v+u\hfill & \text{in}\Omega \times (0,T),\hfill \end{array}$$

in a domain Ω of ${ℝ}^N$ with $N⩾1$, where $m,\Gamma >0$, $\lambda ⩾0$ are constants and $T>0$. When $\Omega \ne {ℝ}^N$, 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.

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