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.

@article{AIHPC_2014__31_1_81_0, author = {Yao, Yao}, title = {Asymptotic behavior for critical {Patlak{\textendash}Keller{\textendash}Segel} model and a repulsive{\textendash}attractive aggregation equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {81--101}, publisher = {Elsevier}, volume = {31}, number = {1}, year = {2014}, doi = {10.1016/j.anihpc.2013.02.002}, mrnumber = {3165280}, zbl = {1288.35094}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.002/} }

TY - JOUR AU - Yao, Yao TI - Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 81 EP - 101 VL - 31 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.002/ DO - 10.1016/j.anihpc.2013.02.002 LA - en ID - AIHPC_2014__31_1_81_0 ER -

%0 Journal Article %A Yao, Yao %T Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 81-101 %V 31 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.002/ %R 10.1016/j.anihpc.2013.02.002 %G en %F AIHPC_2014__31_1_81_0

Yao, Yao. Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 81-101. doi : 10.1016/j.anihpc.2013.02.002. http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.002/

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