The Keller--Segel System on the Two-Dimensional-Hyperbolic Space

Abstract

In this paper, we shall study the parabolic-elliptic Keller--Segel system on the Poincaré disk model of the two-dimensional-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the two-dimensional Euclidean case, under the subcritical 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 hyperbolic 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. 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.