THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC SPACE

Abstract : In this paper, we shall study the parabolic-elliptic Keller-Segel system on the Poincaré disk model of the 2D-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the 2D-Euclidean case, under the sub-critical condition χM < 8π, we shall prove global well-posedness results with any initial L 1-data. More precisely, by using dispersive and smoothing estimates we shall prove Fujita-Kato type theorems for local well-posedness. We shall then use the logarithmic Hardy-Littlewood-Sobolev estimates on the hyper-bolic space to prove that the solution cannot blow-up in finite time. For larger mass χM > 8π, we shall obtain a blow-up result under an additional condition with respect to the flat case, probably due to the spectral gap of the Laplace-Beltrami operator. According to the exponential growth of the hyperbolic space, we find a suitable weighted moment of exponential type on the initial data for blow-up.
Type de document :
Pré-publication, Document de travail
2018
Liste complète des métadonnées

Littérature citée [8 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01899214
Contributeur : Vittoria Pierfelice <>
Soumis le : vendredi 19 octobre 2018 - 13:16:31
Dernière modification le : samedi 20 octobre 2018 - 01:15:21

Fichiers

VPKellerSegelonhyperbolic.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01899214, version 1
  • ARXIV : 1810.08502

Collections

Citation

Patrick Maheux, Vittoria Pierfelice. THE KELLER-SEGEL SYSTEM ON THE 2D-HYPERBOLIC SPACE. 2018. 〈hal-01899214〉

Partager

Métriques

Consultations de la notice

15

Téléchargements de fichiers

7