Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Yao, Yao

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 81-101.

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.

[1] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures Math. ETH Zurich, Birkhäuser, Basel (2008) | MR | Zbl

[2] D. Balagué, J.A. Carrillo, T. Laurent, G. Raoul, Nonlocal interactions by repulsive–attractive potentials: radial ins/stability, arXiv:1109.5258 (2011) | MR | Zbl

[3] J. Bedrossian, I. Kim, Global existence and finite time blow-up for critical Patlak–Keller–Segel models with inhomogeneous diffusion, arXiv:1108.5301 (2011) | MR | Zbl

[4] J. Bedrossian, N. Rodríguez, A.L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion, Nonlinearity 24 (2011), 1683-1714 | MR | Zbl

[5] M. Bertsch, D. Hilhorst, A density dependent diffusion equation in population dynamics: stabilization to equilibrium, SIAM J. Math. Anal. 17 (1986), 863-883 | MR | Zbl

[6] P. Biler, G. Karch, P. Laurençot, T. Nadzieja, The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Models Methods Appl. Sci. 29 (2006), 1563-1583 | MR | Zbl

[7] A. Blanchet, On the parabolic–elliptic Patlak–Keller–Segel system in dimension 2 and higher, arXiv:1109.1543 (2011) | MR

[8] A. Blanchet, J. Carrillo, P. Laurençot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), 133-168 | MR | Zbl

[9] A. Blanchet, J.A. Carrillo, N. Masmoudi, Infinite time aggregation for the critical Patlak–Keller–Segel model in ${ℝ}^2$, Comm. Pure Appl. Math. 61 (2008), 1449-1481 | MR | Zbl

[10] A. Blanchet, J. Dolbeault, B. Perthame, Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 44 (2006), 1-32 | EuDML | MR | Zbl

[11] A. Blanchet, J. Dolbeault, M. Escobedo, J. Fernández, Asymptotic behavior for small mass in the two-dimensional parabolic–elliptic Keller–Segel model, J. Math. Anal. Appl. 361 (2010), 533-542 | MR | Zbl

[12] M. Burger, M. Di Francesco, M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, arXiv:1103.5365 (2011) | MR | Zbl

[13] J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1-82 | MR | Zbl

[14] J.A. Carrillo, R.J. Mccann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 (2006), 217-263 | MR | Zbl

[15] J. Campos, J. Dolbeault, Asymptotic estimates for the parabolic–elliptic Keller–Segel model in the plane, arXiv:1206.1963 (2012) | MR

[16] E. Dibenedetto, Continuity of weak solutions to a general porous media equation, Indiana Univ. Math. J. 32 (1983), 83-118 | MR | Zbl

[17] J. Dolbeault, B. Perthame, Optimal critical mass in the two dimensional Keller–Segel model in ${ℝ}^2$, C. R. Acad. Sci. Paris, Sér. I Math. 339 (2004), 611-616 | MR | Zbl

[18] R.C. Fetecau, Y. Huang, Equilibria of biological aggregations with nonlocal repulsive–attractive interactions, arXiv:1109.2864 (2011) | MR | Zbl

[19] R.C. Fetecau, Y. Huang, T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity 24 (2011), 2681-2716 | MR | Zbl

[20] I. Gallagher, Th. Gallay, P.-L. Lions, 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 | Zbl

[21] W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824 | MR | Zbl

[22] E.F. Keller, L.A. Segel, Model for chemotaxis, J. Theoret. Biol. 30 (1971), 225-234 | Zbl

[23] I. Kim, Y. Yao, 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 | Zbl

[24] T. Kolokolnikov, H. Sun, D. Uminsky, A.L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Commun. 84 (2011), 015203(R)

[25] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103-165 | MR | Zbl

[26] A.J. Leverentz, C.M. Topaz, Andrew J. Bernoff, Asymptotic dynamics of attractive–repulsive swarms, SIAM J. Appl. Dyn. Syst. 8 (2009), 880-908 | MR | Zbl

[27] A. Mogilner, L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol. 38 (1999), 534-570 | MR | Zbl

[28] C.S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338 | MR | Zbl

[29] T. Senba, T. Suzuki, Weak solutions to a parabolic–elliptic system of chemotaxis, J. Funct. Anal. 47 (2001), 17-51 | MR | Zbl

[30] C.M. Topaz, A.L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), 152-174 | MR | Zbl

[31] C.M. Topaz, A.L. Bertozzi, M.A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), 1601-1623 | MR | Zbl

[32] J. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press (2007) | MR | Zbl

[33] C. Villani, 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 | Zbl