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 , 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--Keller--Segel model and a repulsive--attractive aggregation equation}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {31}, year = {2014}, pages = {81-101}, doi = {10.1016/j.anihpc.2013.02.002}, mrnumber = {3165280}, zbl = {1288.35094}, language = {en}, url = {http://dml.mathdoc.fr/item/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, Tome 31 (2014) pp. 81-101. doi : 10.1016/j.anihpc.2013.02.002. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_1_81_0/
[1] Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zurich, Birkhäuser, Basel (2008) | MR 2401600 | Zbl 1145.35001
, , ,[2] Nonlocal interactions by repulsive–attractive potentials: radial ins/stability, arXiv:1109.5258 (2011) | MR 3143991 | Zbl 1286.35038
, , , ,[3] Global existence and finite time blow-up for critical Patlak–Keller–Segel models with inhomogeneous diffusion, arXiv:1108.5301 (2011) | MR 3048211 | Zbl 1294.35172
, ,[4] Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion, Nonlinearity 24 (2011), 1683-1714 | MR 2793895 | Zbl 1222.49038
, , ,[5] A density dependent diffusion equation in population dynamics: stabilization to equilibrium, SIAM J. Math. Anal. 17 (1986), 863-883 | MR 846394 | Zbl 0607.35052
, ,[6] The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Models Methods Appl. Sci. 29 (2006), 1563-1583 | MR 2249579 | Zbl 1105.35131
, , , ,[7] On the parabolic–elliptic Patlak–Keller–Segel system in dimension 2 and higher, arXiv:1109.1543 (2011) | MR 3379847
,[8] Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), 133-168 | MR 2481820 | Zbl 1172.35035
, , ,[9] Infinite time aggregation for the critical Patlak–Keller–Segel model in , Comm. Pure Appl. Math. 61 (2008), 1449-1481 | MR 2436186 | Zbl 1155.35100
, , ,[10] Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 44 (2006), 1-32 | MR 2226917 | Zbl 1112.35023
, , ,[11] Asymptotic behavior for small mass in the two-dimensional parabolic–elliptic Keller–Segel model, J. Math. Anal. Appl. 361 (2010), 533-542 | MR 2568716 | Zbl 1176.35029
, , , ,[12] Stationary states of quadratic diffusion equations with long-range attraction, arXiv:1103.5365 (2011) | MR 3061139 | Zbl 1282.35203
, , ,[13] Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1-82 | MR 1853037 | Zbl 0984.35027
, , , , ,[14] Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 (2006), 217-263 | MR 2209130 | Zbl 1082.76105
, , ,[15] Asymptotic estimates for the parabolic–elliptic Keller–Segel model in the plane, arXiv:1206.1963 (2012) | MR 3196188
, ,[16] Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J. 32 (1983), 83-118 | MR 684758 | Zbl 0526.35042
,[17] Optimal critical mass in the two dimensional Keller–Segel model in , C. R. Acad. Sci. Paris, Sér. I Math. 339 (2004), 611-616 | MR 2103197 | Zbl 1056.35076
, ,[18] Equilibria of biological aggregations with nonlocal repulsive–attractive interactions, arXiv:1109.2864 (2011) | MR 3143993 | Zbl 1286.35017
, ,[19] Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity 24 (2011), 2681-2716 | MR 2834242 | Zbl 1288.92031
, , ,[20] On the uniqueness of the solution of the two-dimensional Navier–Stokes equation with a Dirac mass as initial vorticity, Math. Nachr. 278 (2005), 1665-1672 | MR 2176270 | Zbl 1083.35092
, , ,[21] On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824 | MR 1046835 | Zbl 0746.35002
, ,[22] Model for chemotaxis, J. Theoret. Biol. 30 (1971), 225-234 | Zbl 1170.92307
, ,[23] The Patlak–Keller–Segel model and its variations: properties of solutions via maximum principle, SIAM J. Math. Anal. 44 (2012), 568-602, arXiv:1102.0092 | MR 2914242 | Zbl 1261.35080
, ,[24] A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Commun. 84 (2011), 015203(R)
, , , ,[25] From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103-165 | MR 2013508 | Zbl 1071.35001
,[26] Asymptotic dynamics of attractive–repulsive swarms, SIAM J. Appl. Dyn. Syst. 8 (2009), 880-908 | MR 2533628 | Zbl 1168.92045
, , ,[27] A non-local model for a swarm, J. Math. Biol. 38 (1999), 534-570 | MR 1698215 | Zbl 0940.92032
, ,[28] Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338 | MR 81586 | Zbl 1296.82044
,[29] Weak solutions to a parabolic–elliptic system of chemotaxis, J. Funct. Anal. 47 (2001), 17-51 | MR 1909263 | Zbl 1005.35026
, ,[30] Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), 152-174 | MR 2111591 | Zbl 1071.92048
, ,[31] A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), 1601-1623 | MR 2257718 | Zbl 1334.92468
, , ,[32] The Porous Medium Equation: Mathematical Theory, Oxford University Press (2007) | MR 2286292 | Zbl 1107.35003
,[33] Optimal transportation, dissipative PDEs and functional inequalities, Optimal Transportation and Applications, Martina Franca, 2001, Lecture Notes in Math. vol. 1813, Springer, Berlin (2003), 53-89 | MR 2006305 | Zbl 1039.35147
,