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Preprints, Working Papers, ... Year : 2018

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

Patrick Maheux
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  • PersonId : 832896
Vittoria Pierfelice
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  • PersonId : 846244

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.
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Dates and versions

hal-01899214 , version 1 (19-10-2018)

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