On blowup dynamics in the Keller--Segel model of chemotaxis

S. I. Dejak ; D. Egli ; P. M. Lushnikov ; I. M. Sigal

Algebra i analiz, Tome 25 (2013) no. 4, pp. 47-84.

Résumé

The (reduced) Keller–Segel equations modeling chemotaxis of bio-organisms are investigated. A formal derivation and partial rigorous results of the blowup dynamics are presented for solutions of these equations describing the chemotactic aggregation of the organisms. The results are confirmed by numerical simulations, and the formula derived coincides with the formula of Herrero and Velázquez for specially constructed solutions.

Détail

Keywords: reaction-diffusion equations, nonlinear partial differential equations, blowup, collapse, chemotaxis, Keller–Segel equation, blowup profile.

Bibliographie

[1] Alber M., Chen N., Glimm T., Lushnikov P. M., “Multiscale dynamics of biological cells with chemotaktic interactions: from a discrete stochastic model to a continuous description”, Phys. Rev. E (3), 73:5 (2006), 051901, 11 pp. | DOI | MR

[2] Alber M., Chen N., Lushnikov P. M., Newman S. A., “Continuous macroscopic limit of a discrete stochastic model for interaction of living cells”, Phys. Rev. Lett., 99 (2007), 168102 | DOI

[3] Beckner W., “Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality”, Ann. of Math. (2), 138:1 (1993), 213–242 | DOI | MR | Zbl

[4] Bertozzi A. L., Carillo J. A., Laurent Th., “Blow-up in multidimensional aggregation equations with mildly singular interaction kernels”, Nonlinearity, 22:3 (2009), 683–710 | DOI | MR | Zbl

[5] Biler P., “Growth and accretion of mass in an astrophysical model”, Appl. Math. (Warsaw), 23:2 (1995), 179–189 | MR | Zbl

[6] Biler P., “Local and global solvability of some parabolic systems modeling chemotaxis”, Adv. Math. Sci. Appl., 8:2 (1998), 715–743 | MR | Zbl

[7] Biler P., Karch G., Laurencot P., The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane, Preprint, 2006 | MR

[8] Bizoń P., Ovchinnikov Yu. N., Sigal I. M., “Collapse of an instanton”, Nonlinearity, 17:4 (2004), 1179–1191 | DOI | MR | Zbl

[9] Blanchet A., Carlen E., Carrillo J. A., “Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model”, J. Funct. Anal., 262:5 (2012), 2142–2230 ; arXiv: 1009.0134 | DOI | MR | Zbl

[10] Blanchet A., Carrillo J. A., Laurençot P., “Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions”, Calc. Var. Partial Diff. Equations, 35:2 (2009), 133–168 | DOI | MR | Zbl

[11] Blanchet A., Carrillo J. A., Masmoudi N., “Infinite time aggregation for the critical Patlak–Keller–Segel model in $\mathbb R^2$”, Comm. Pure Appl. Math., 61:10 (2008), 1449–1481 | DOI | MR | Zbl

[12] Blanchet A., Dolbeault J., Escobedo M., Fernandez J., “Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller–Segel model”, J. Math. Anal. Appl., 361:2 (2010), 533–542 | DOI | MR | Zbl

[13] Blanchet A., Dolbeault J., Perthame B., “Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions”, Electron. J. Diff. Equations, 2006 (2006), No 44, 32 pp., (electronic) | MR

[14] Bonner J. T., The cellular slime molds, Princeton Univ. Press, Princeton, NJ, 1967

[15] Brenner M. P., Constantin P., Kadanoff L. P., Schenkel A., Venkataramani S. C., “Diffusion, attraction and collapse”, Nonlinearity, 12:4 (1999), 1071–1098 | DOI | MR | Zbl

[16] Brenner M. P., Levitov L. S., Budrene E. O., “Physical mechanisms for chemotactic pattern formation by bacteria”, Biophys. J., 74 (1998), 1677–1693 | DOI

[17] Buslaev V. S., Perel'man G. S., “On the stability of solitary waves for nonlinear Schrödinger equations”, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 75–98 | MR | Zbl

[18] Buslaev V. S., Sulem C., “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20:3 (2003), 419–475 | DOI | MR | Zbl

[19] Carlen E., Loss M., “Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on Sn”, Geom. Funct. Anal., 2:2 (1992), 90–104 | DOI | MR | Zbl

[20] Carlen E., Figalli A., Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation, arXiv: 1107.5976 | MR

[21] Carmeliet P., “Mechanisms of angiogenesis and arteriogenesis”, Nat. Med., 6 (2000), 389–395 | DOI

[22] Carrillo J. A., Fornasier M., Toscani G., Vecil F., “Particle, kinetic, and hydrodynamic models of swarming”, Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010, 297–336 | MR | Zbl

[23] Chavanis P.-H., Sire C., “Exact analytical solution of the collapse of self-gravitating Brownian particles and bacterial populations at zero temperature”, Phys. Rev. E (3), 83:3 (2011), 031131, 20 pp. | DOI | MR

[24] Childress S., Percuss J. K., “Nonlinear aspects of chemotaxis”, Math. Bisosc., 56:3–4 (1981), 217–237 | DOI | MR | Zbl

[25] Constantin P., Kevrekidis I. G., Titi E. S., “Asymptotic states of a Smoluchowski equation”, Arch. Ration. Mech. Anal., 174:3 (2004), 365–384 | DOI | MR | Zbl

[26] Constantin P., Kevrekidis I., Titi E. S., “Remarks on a Smoluchowski equation”, Discrete Contin. Dyn. Syst., 11:1 (2004), 101–112 | DOI | MR | Zbl

[27] Cycon H. L., Froese R. G., Kirsch W., Simon B., Schrödinger Operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics. Springer study edition, Springer-Verlag, Berlin, 1987 | MR | Zbl

[28] Dejak S., Gang Z., Sigal I. M., Wang S., “Blow-up dynamics in nonlinear heat equations”, Adv. in Appl. Math., 40:4 (2008), 433–481 | DOI | MR | Zbl

[29] Dejak S. I., Lushnikov P. M., Ovchinnikov Yu. N., Sigal I. M., “On spectra of linearized operators for Keller–Segel models of chemotaxis”, Phys. D, 241:15 (2012), 1245–1254 | DOI | MR | Zbl

[30] Doi M., “Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases”, J. Polym. Sci., Polym. Phys. Ed., 19 (1981), 229–243 | DOI

[31] Dyachenko S. A., Lushnikov P. M., Vladimirova N., “Logarithmic-type Scaling of the Collapse of Keller–Segel Equation”, Numerical Analysis and Applied Mathematics, ICNAAM 2011, AIP Conf. Proc., 1389, 2011, 709–712 | DOI

[32] Gajewski H., Zacharias K., “Global behavior of a reaction-diffusion system modelling chemotaxis”, Math. Nachr., 195 (1998), 77–114 | DOI | MR | Zbl

[33] Gang Zhou, Sigal I. M., “On soliton dynamics in nonlinear Schrödinger equations”, Geom. Funct. Anal., 16:6 (2006), 1377–1390 | DOI | MR | Zbl

[34] Gang Zhou, Sigal I. M., “Relaxation of solitons in nonlinear Schrödinger equations with potentials”, Adv. Math., 216:2 (2007), 443–490 | DOI | MR | Zbl

[35] Gustafson S., Sigal I. M., Mathematical concepts of quantum mechanics, Springer-Verlag, Heidelberg, 2011 | MR | Zbl

[36] Herrero M. A., Velázquez J. J. L., “Chemptactic collapse for the Keller–Segel model”, J. Math. Biol., 35:2 (1996), 177–194 | DOI | MR | Zbl

[37] Herrero M. A., Velázquez J. J. L., “Singularity patterns in a chemotaxis model”, Math. Ann., 306:3 (1996), 583–623 | DOI | MR | Zbl

[38] Herrero M. A., Velázquez J. J. L., “A blow-up mechanism for a chemotaxis model”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633–683 | MR | Zbl

[39] Herrero M. A., Medina E., Velázquez J. J. L., “Finite-time aggregation into a single point in a reaction-diffusion system”, Nonlinearity, 10 (1997), 1739–1754 | DOI | MR | Zbl

[40] Herrero M. A., Medina E., Velázquez J. J. L., “Self-similar blow-up for a reaction-diffusion system”, J. Comput. Appl. Math., 97:1–2 (1998), 99–119 | DOI | MR | Zbl

[41] Hillen T., Painter K. J., “A user's guide to PDE models for chemotaxis”, J. Math. Biol., 58:1–2 (2009), 183–217 | DOI | MR | Zbl

[42] Horstmann D., “The nonsymmetric case of the Keller–Segel model in chemotaxis: some recent results”, NoDEA Nonlinear Differential Equations Appl., 8:4 (2001), 399–423 | DOI | MR | Zbl

[43] Horstmann D., “On the existence of radially symmetric blow-up solutions for the Keller–Segel model”, J. Math. Biol., 44:5 (2002), 463–478 | DOI | MR | Zbl

[44] Horstmann D., “From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. I”, Jahresber. Deutsch. Math.-Verein, 105:3 (2003), 103–165 | MR | Zbl

[45] Horstmann D., “From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. II”, Jahresber. Deutsch. Math.-Verein, 106:2 (2004), 51–69 | MR | Zbl

[46] Horstmann D., Wang G., “Blowup in a chemotaxis model without symmetry assumptions”, European J. Appl. Math., 12:2 (2001), 159–177 | DOI | MR | Zbl

[47] Horstmann D., Winkler M., “Boundedness vs. blowup in a chemotaxis system”, J. Differential Equations, 215:1 (2005), 52–107 | DOI | MR | Zbl

[48] Jäger W., Luckhaus S., “On explosions of solutions to a system of partial differential equations modelling chemotaxis”, Trans. Amer. Soc., 329:2 (1992), 819–824 | MR | Zbl

[49] Keller E. F., Segel L. A., “Initiation of slime mold aggregation viewed as an instability”, J. Theor. Biol., 26 (1970), 300–415 | DOI

[50] Krieger J., Schlag W., Tataru D., “Renormalization and blow up for charge one equivariant critical wave maps”, Invent. Math., 171:3 (2008), 543–615 | DOI | MR | Zbl

[51] Krieger J., Schlag W., Tataru D., Renormalization and blow up for critical Yang–Mills problem, 2008, arXiv: 0809.2114 | MR

[52] Larson R. G., The structure and rheology of complex fluids, Oxford Univ. Press, London, 1999

[53] Lushnikov P. M., “Critical chemotactic collapse”, Phys. Lett. A, 374 (2010), 1678–1685 | DOI | Zbl

[54] Lushnikov P. M., Chen N., Alber M., “Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact”, Phys. Rev. E, 78:6 (2008), 061904, 12 pp. | DOI

[55] Merle F., Rafael P., “Blow-up of the critical norm for some radial $L2$ super critical nonlinear Schrödinger equations”, Amer. J. Math., 130:4 (2008), 945–978 | DOI | MR | Zbl

[56] Merle F., Zaag H., “Blow-up behavior outside the origin for a semilinear wave equation in the radial case”, Bull. Sci. Math., 135:4 (2011), 353–373 | DOI | MR | Zbl

[57] Nanjundiah V., “Chemotaxis, signal relaying and aggregation morphology”, J. Theor. Biol., 42 (1973), 63–105 | DOI

[58] Nagai T., “Blow-up of radially symmetric solutions to a chemotaxis system”, Adv. Math. Sci. Appl., 5:2 (1995), 581–601 | MR | Zbl

[59] Nagai T., “Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains”, J. Inequal. Appl., 6:1 (2001), 37–55 | MR | Zbl

[60] Nagai T., “Global existence and blowup of solutions to a chemotaxis system”, Proc. of the Third World Congress of Nonlinear Analysis, Part 2 (Catania, 2000), Nonlinear Anal., 47, no. 2, 2001, 777–787 | DOI | MR | Zbl

[61] Nagai T., Senba T., “Behavior of radially symmetric solutions of a system related to chemotaxis”, Proc. of the Third World Congress of Nonlinear Analysis, Part 6 (Athens, 1996), Nonlinear Anal., 30, no. 6, 1997, 3837–3842 | DOI | MR | Zbl

[62] Nagai T., Senba T., “Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis”, Adv. Math. Sci. Appl., 8:1 (1998), 145–146 | MR

[63] Nagai T., Senba T., Suzuki T., “Chemotactic collapse in a parabolic system of mathematical biology”, Hiroshima Math. J., 30:3 (2000), 463–497 | MR | Zbl

[64] Nagai T., Senba T., Yoshida K., “Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis”, Funkcial. Ekvac., 40:3 (1997), 411–433 | MR | Zbl

[65] Newman T. J., Grima R., “Many-body theory of chemotactic cell-cell interactions”, Phys. Rev. E, 70:5 (2004), 051916, 15 pp. | DOI

[66] Oelschläger K., “On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes”, Probab. Theory Related Fields, 82:4 (1989), 565–586 | DOI | MR | Zbl

[67] Ostriker J., “The equilibrium of polytropic and isotermal cylinders”, Astrophys. J., 140 (1964), 1056–1066 | DOI | MR

[68] Othmer H. G., Stevens A., “Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks”, SIAM J. Appl. Math., 57:4 (1997), 1044–1081 | DOI | MR | Zbl

[69] Ovchinnikov Yu. N., Sigal I. M., “On collapse of wave maps”, Physica D, 240:17 (2011), 1311–1324 | DOI | MR | Zbl

[70] Patlak C. S., “Random walk with persistence and external bias”, Bull. Math. Biophys., 15 (1953), 311–338 | DOI | MR | Zbl

[71] Perthame B., Transport equations in biology, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007 | MR | Zbl

[72] Rafael P., Rodnianski I., Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems, 2010, arXiv: 0911.0692

[73] Reed M., Simon B., Methods of modern mathematical physics, v. IV, Analysis of operators, Academic Press, New–York–London, 1978 | MR | Zbl

[74] Rodnianski I., Sternbenz J., “On the formation of singularities in the critical $O(3)\sigma$-model”, Ann. of Math. (2), 172:1 (2010), 187–242 | DOI | MR | Zbl

[75] Sire C., Chavanis P.-H., “Critical dynamics of self-gravitating Langevin particles and bacterial populations”, Phys. Rev. E (3), 78:6 (2008), 061111, 22 pp. | DOI | MR

[76] Soffer A., Weinstein I. M., “Multichannel nonlinear scattering for nonintegrable equations”, Comm. Math. Phys., 133:1 (1990), 119–146 | DOI | MR | Zbl

[77] Soffer A., Weinstein I. M., “Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data”, J. Diff. Equations, 98:2 (1992), 376–390 | DOI | MR | Zbl

[78] Soffer A., Weinstein I. M., “Selection of the ground state for nonlinear Schrödinger equations”, Rev. Math. Phys., 16:8 (2004), 977–1071 ; (2003), arXiv: nlin.PS/0308020 | DOI | MR | Zbl

[79] Stevens A., “The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems”, SIAM J. Appl. Math., 61:1 (2000), 183–212, (electronic) | DOI | MR | Zbl

[80] Struwe M., “Equivariant wave maps in two space dimensions”, Dedicated to the memory of Sürgen K. Moser, Comm. Pure Appl. Math., 56:7 (2003), 815–823 | DOI | MR | Zbl

[81] Sulem C., Sulem P. L., Nonlinear Schrödinger Equation, Self-focusing and wave collapse, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999 | MR | Zbl

[82] Tsai T.-P., Yau H.-T., “Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions”, Comm. Pure Appl. Math., 55:2 (2002), 153–216 | DOI | MR | Zbl

[83] Tsai T.-P., Yau H.-T., “Relaxation of excited states in nonlinear Schrödinger Equations”, Int. Math. Res. Not., 2002:31 (2002), 1629–1673 | DOI | MR | Zbl

[84] Tsai T.-P., Yau H.-T., “Stable directions for excited states of nonlinear Schrödinger equations”, Comm. Partial Differential Equations, 27:11–12 (2002), 2363–2402 | DOI | MR | Zbl

[85] Velázquez J. J. L., “Stability of some mechanisms of chemotactic aggregation”, SIAM J. Appl. Math., 62:5 (2002), 1581–1633 | DOI | MR | Zbl

[86] Velázquez J. J. L., “Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions”, SIAM J. Appl. Math., 64:4 (2004), 1198–1248 | DOI | MR

[87] Wolansky G., “On steady distributions of self-attracting clusters under friction and fluctuations”, Arch. Rational Mech. Anal., 119:4 (1992), 355–391 | DOI | MR | Zbl