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Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane

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Date
2014
Dewey
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Sujet
Lyapunov functiona; spectral gap; relative entropy; free energy; self-similar solutions; logarithmic Hardy-Littlewood-Sobolev inequality; subcritical mass; large time asymptotics; chemotaxis; Keller-Segel model
Journal issue
Communications in Partial Differential Equations
Volume
39
Number
5
Publication date
2014
Article pages
806-841
Publisher
Marcel Dekker
DOI
http://dx.doi.org/10.1080/03605302.2014.885046
URI
https://basepub.dauphine.fr/handle/123456789/11954
Collections
  • CEREMADE : Publications
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Author
Campos Serrano, Juan
Dolbeault, Jean
Type
Article accepté pour publication ou publié
Abstract (EN)
We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8π after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of J.I. Diaz, T. Nagai, and J.-M. Rakotoson, and allow us to establish uniform estimates for Lp norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest.

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