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Infinite time aggregation for the critical Patlak-Keller-Segel model in $$\mathbb R^2$$. (English) Zbl 1155.35100
For the two-dimensional Smoluchowski-Poisson equation (also called the parabolic-elliptic Keller-Segel chemotaxis model)
\begin{aligned} \partial_t n &= \nabla\cdot (\nabla n-\chi n \nabla c), \quad (t,x)\in (0,\infty)\times{\mathbb R}^2,\\ c(t,x)&=-\frac{1}{2\pi} \int_{{\mathbb R}^2} \ln| x-y| n(t,y)\,dy, \quad (t,x)\in (0,\infty)\times{\mathbb R}^2,\\ n(0,x)& = n_0(x)\geq 0, \quad x\in {\mathbb R}^2, \end{aligned} the value of the $$L^1$$-norm of $$n_0$$ governs the maximal existence time of the solution $$n$$. More precisely, given $$\chi>0$$, if $$\chi \| n_0\| _1>8\pi$$, then the solution becomes unbounded in finite time while the solution is global and bounded if $$\chi \| n_0\| _1<8\pi$$. The critical case $$\chi \| n_0\| _1=8\pi$$ is analysed in the paper under review: the solution is proved to be global and becomes unbounded as $$t\to\infty$$. More precisely, $$n(t)$$ behaves as $$(8\pi/\chi)\delta_{x_M}$$ as $$t\to\infty$$, $$x_M$$ denoting the centre of mass of $$n_0$$ and $$\delta_{x_M}$$ the Dirac measure centred at $$x_M$$.

##### MSC:
 35Q80 Applications of PDE in areas other than physics (MSC2000) 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 92C17 Cell movement (chemotaxis, etc.)
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##### References:
 [1] Arnold, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh Math 142 (1) pp 35– (2004) · Zbl 1063.35109 [2] Aubin, Un théorème de compacité, C R Acad Sci Paris 256 pp 5042– (1963) [3] Bavaud, Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation, Rev Modern Phys 63 (1) pp 129– (1991) [4] Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann of Math (2) 138 (1) pp 213– (1993) · Zbl 0826.58042 [5] Benguria, R. D. Von Weizsaecker and exchange corrections in the Thomas Fermi theory. Doctoral dissertation, Princeton University, Princeton, N.J., 1979. [6] Benguria, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm Math Phys 79 (2) pp 167– (1981) · Zbl 0478.49035 [7] Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv Math Sci Appl 8 (2) pp 715– (1998) · Zbl 0913.35021 [8] Biler, The 8$$\pi$$-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol Methods Nonlinear Anal 27 (1) pp 133– (2006) · Zbl 1135.35367 [9] Biler, The 8$$\pi$$-problem for radially symmetric solutions of a chemotaxis model in the plane, Math Methods Appl Sci 29 (13) pp 1563– (2006) · Zbl 1105.35131 [10] Biler, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq Math 66 (1) pp 131– (1993) · Zbl 0818.35046 [11] Biler, Global and exploding solutions in a model of self-gravitating systems, Rep Math Phys 52 (2) pp 205– (2003) · Zbl 1043.85001 [12] Billingsley, Convergence of probability measures (1999) · Zbl 0944.60003 · doi:10.1002/9780470316962 [13] Blanchet, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron J Differential Equations 2006 (44) pp 1– · Zbl 1112.35023 [14] Calvez, Volume effects in the Keller-Segel model: energy estimates preventing blow-up, J Math Pures Appl (9) 86 (2) pp 155– (2006) · Zbl 1116.35057 · doi:10.1016/j.matpur.2006.04.002 [15] Calvez, V.; Perthame, B.; Sharifi Tabar, M. Modified Keller-Segel system and critical mass for the log interaction kernel. preprint 2006. Available online at: http://calvino.polito.it/mcrtnline/library.html. · Zbl 1126.35077 [16] Carlen, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on Sn, Geom Funct Anal 2 (1) pp 90– (1992) · Zbl 0754.47041 [17] Chalub, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh Math 142 (1) pp 123– (2004) · Zbl 1052.92005 [18] Chen, Classification of solutions of some nonlinear elliptic equations, Duke Math J 63 (3) pp 615– (1991) [19] Childress, Nonlinear aspects of chemotaxis, Math Biosci 56 (3) pp 217– (1981) · Zbl 0481.92010 [20] Corrias, A chemotaxis model motivated by angiogenesis, C R Math Acad Sci Paris 336 (2) pp 141– (2003) · Zbl 1028.35062 · doi:10.1016/S1631-073X(02)00008-0 [21] Corrias, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J Math 72 pp 1– (2004) · Zbl 1115.35136 [22] Dolbeault, Optimal critical mass in the two-dimensional Keller-Segel model in $$\mathbb{R}$$2, C R Math Acad Sci Paris 339 (9) pp 611– (2004) · Zbl 1056.35076 · doi:10.1016/j.crma.2004.08.011 [23] Gabetta, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J Statist Phys 81 (5) pp 901– (1995) · Zbl 1081.82616 [24] Gajewski, Global behavior of a reaction-diffusion system modelling chemotaxis, Math Nachr 195 pp 77– (1998) · Zbl 0918.35064 · doi:10.1002/mana.19981950106 [25] Herrero, Singularity patterns in a chemotaxis model, Math Ann 306 (3) pp 583– (1996) · Zbl 0864.35008 [26] Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber Deutsch Math-Verein 105 (3) pp 103– (2003) · Zbl 1071.35001 [27] Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber Deutsch Math-Verein 106 (2) pp 51– (2004) · Zbl 1072.35007 [28] Horstmann, Blow-up in a chemotaxis model without symmetry assumptions, European J Appl Math 12 (2) pp 159– (2001) · Zbl 1017.92006 [29] Jäger, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans Amer Math Soc 329 (2) pp 819– (1992) · Zbl 0746.35002 [30] Keller, Initiation of slide mold aggregation viewed as an instability, J Theor Biol 26 pp 399– (1970) · Zbl 1170.92306 [31] Kowalczyk, Preventing blow-up in a chemotaxis model, J Math Anal Appl 305 (2) pp 566– (2005) · Zbl 1065.35063 [32] Laurençot, P. Personal communication, 2006. [33] Laurençot, The continuous coagulation-fragmentation equations with diffusion, Arch Ration Mech Anal 162 (1) pp 45– (2002) · Zbl 0997.45005 [34] Lê Châu-Hoàn. Etude de la classe des opérateurs m-accrétifs de L1($$\Omega$$) et accrétifs dans L($$\Omega$$). Thèse de 3ème cycle, Université de Paris VI, 1977. [35] Lions, Équations différentielles opérationnelles et problèmes aux limites 111 (1961) [36] Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixed finite elements, M2AN Math Model Numer Anal 37 (4) pp 617– (2003) · Zbl 1065.92006 [37] Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv Math Sci Appl 5 (2) pp 581– (1995) · Zbl 0843.92007 [38] Nagai, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial Ekvac 40 (3) pp 411– (1997) · Zbl 0901.35104 [39] Naito, Self-similar solutions to a nonlinear parabolic-elliptic system. Proceedings of Third East Asia Partial Differential Equation Conference, Taiwanese J Math 8 (1) pp 43– (2004) · Zbl 1113.35058 [40] Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J Theor Biol 42 (1) pp 63– (1973) [41] Othmer, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks, SIAM J Appl Math 57 (4) pp 1044– (1997) · Zbl 0990.35128 [42] Padmanabhan, Statistical mechanics of gravitating systems, Phys Rep 188 (5) pp 285– (1990) · Zbl 1211.82001 [43] Patlak, Random walk with persistence and external bias, Bull Math Biophys 15 (3) pp 311– (1953) · Zbl 1296.82044 [44] Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic, Appl Math 49 (6) pp 539– (2004) · Zbl 1099.35157 [45] Senba, Weak solutions to a parabolic-elliptic system of chemotaxis, J Funct Anal 191 (1) pp 17– (2002) · Zbl 1005.35026 [46] Senba, Applied analysis. Mathematical methods in natural science (2004) · Zbl 1053.00001 · doi:10.1142/p320 [47] Simon, Compact sets in the space Lp(0, T; B), Ann Mat Pura Appl (4) 146 pp 65– (1987) [48] Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J Appl Math 61 (1) pp 183– (2000) · Zbl 0963.60093 [49] Suzuki, Free energy and self-interacting particles 62 (2005) · Zbl 1082.35006 · doi:10.1007/0-8176-4436-9 [50] van Duijn, Global existence conditions for a nonlocal problem arising in statistical mechanics, Adv Differential Equations 9 (1) pp 133– (2004) · Zbl 1108.35137 [51] Velázquez, Stability of some mechanisms of chemotactic aggregation, SIAM J Appl Math 62 (5) pp 1581– (2002) · Zbl 1013.35004 [52] Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J Appl Math 64 (4) pp 1198– (2004) · Zbl 1058.35021 [53] Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J Appl Math 64 (4) pp 1224– (2004) · Zbl 1058.35022 [54] Wolansky, Comparison between two models of self-gravitating clusters: conditions for gravitational collapse, Nonlinear Anal 24 (7) pp 1119– (1995) · Zbl 0858.70006
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