Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane
Campos Serrano, Juan; Dolbeault, Jean (2014), Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane, Communications in Partial Differential Equations, 39, 5, p. 806-841. http://dx.doi.org/10.1080/03605302.2014.885046
TypeArticle accepté pour publication ou publié
Journal nameCommunications in Partial Differential Equations
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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.
Subjects / KeywordsLyapunov functiona; spectral gap; relative entropy; free energy; self-similar solutions; logarithmic Hardy-Littlewood-Sobolev inequality; subcritical mass; large time asymptotics; chemotaxis; Keller-Segel model
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