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In this paper we study the long time asymptotic behavior for a class ofdiffusion-aggregation equations. Most results except the ones in Section 3.3concern radial solutions. The main tools used in the paper aremaximum-principle type arguments on mass concentration of solutions, as well asenergy method. For the Patlak-Keller-Segel problem with critical power$m=2-2/d$, we prove that all radial solutions with critical mass would convergeto a family of stationary solutions, while all radial solutions withsubcritical mass converge to a self-similar dissipating solution algebraicallyfast. For non-radial solutions, we obtain convergence towards the self-similardissipating solution when the mass is sufficiently small. We also apply themass comparison method to another aggregation model with repulsive-attractiveinteraction, and prove that radial solutions converge to the stationarysolution exponentially fast.

Asymptotic Behavior for Critical Patlak-Keller-Segel model and an Repulsive-Attractive Aggregation Equation