Communications in Mathematical Sciences

Volume 9 (2011)

Number 4

Intermediate asymptotics for critical and supercritical aggregation equations and patlak-keller-segel models

Pages: 1143 – 1161

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n4.a11

Author

Jacob Bedrossian (Department of Mathematics, University of California at Los Angeles)

Abstract

We examine the long-term asymptotic behavior of dissipating solutions to aggregation equations and Patlak-Keller-Segel models with degenerate power-law and linear diffusion. The purpose of this work is to identify when solutions decay to the self-similar spreading solutions of the homogeneous diffusion equations. Combined with strong decay estimates, entropy-entropy dissipation methods provide a natural solution to this question and make it possible to derive quantitative convergence rates in L¹. The estimated rate depends only on the nonlinearity of the diffusion and the strength of the interaction kernel at long range.

Keywords

Patlak-Keller-Segel, aggregation equations, nonlinear diffusion, nonlocal PDE

2010 Mathematics Subject Classification

35B40, 35B45, 35Q92

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