Infinite time blow-up in the Keller-Segel system: existence and stability

Juan Davila (1) , Manuel del Pino (1) , Jean Dolbeault (2) , Monica Musso (1, 3) , Juncheng Wei (4)

1 University of Bath [Bath]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Departamento de Matemáticas [Santiago de Chile]
4 UBC - University of British Columbia

The simplest version of the parabolic-elliptic Patlak-Keller-Segel system in the two-dimensional Euclidean space has an 8π critical mass which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. Among functions with mass 8π, we find a neighborhood of a radial function such that any solution with initial condition in this neighborhood is globally defined and blows-up in infinite time with an explicit scaling involving the square root of the logarithm of the time.

Mots clés: Patlak-Keller-Segel system; chemotaxis; critical mass; blow-up; infinite time blow-up; inner-outer gluing scheme; rate; blow-up profile

Domaines: Equations aux dérivées partielles [math.AP]

Juan Davila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei. Infinite time blow-up in the Keller-Segel system: existence and stability. 2019. ⟨hal-02394787v1⟩