The Keller-segel System On The 2d-hyperbolic Space

TL;DR

This paper studies the Keller-Segel system on the 2D hyperbolic space, revealing how negative curvature affects solution behavior, including conditions for global existence and finite-time blow-up.

Contribution

It extends Keller-Segel analysis to hyperbolic geometry, proving global well-posedness under sub-critical mass and establishing blow-up criteria for larger mass.

Findings

Global well-posedness for sub-critical mass using dispersive estimates

Finite-time blow-up for super-critical mass under certain conditions

Influence of hyperbolic geometry on solution dynamics and blow-up behavior

Abstract

In this paper, we shall study the parabolic-elliptic Keller-Segel system on the Poincar{\'e} 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 $\chi$M < 8$\pi$, 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 $\chi$M > 8$\pi$, 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.