Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation
Abstract
In this paper we study the long time asymptotic behavior for a class of diffusion-aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum-principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak-Keller-Segel problem with critical power $m=2-2/d$, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive-attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2011
- DOI:
- 10.48550/arXiv.1112.4617
- arXiv:
- arXiv:1112.4617
- Bibcode:
- 2011arXiv1112.4617Y
- Keywords:
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- Mathematics - Analysis of PDEs