CONSTRUCTION OF TYPE I-LOG BLOWUP FOR THE KELLER-SEGEL SYSTEM IN DIMENSIONS 3 AND 4
Résumé
We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system
$$\partial_t u = \Delta u - \nabla \cdot (u \nabla K_u), \quad -\Delta K_u = u \quad \mbox{in}\;\; \mathbb{R}b^d,\; d = 3,4,$$
and derive the final blowup profile
$$ u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \mbox{as}\;\; r \to 0, \;\; c_d > 0.$$
To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and
Venkataramani (Nonlinearity, 1999).
Domaines
Mathématiques [math]| Origine | Fichiers produits par l'(les) auteur(s) |
|---|