Abstract
The paper contains results on the asymptotic behavior, as t → +∞, of small solutions to simplified Keller–Segel problem modeling chemotaxis in the whole space \({\mathbb R^2}\). We prove that the multiple of the heat kernel is a surprisingly good approximation of solutions.
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References
Biler P.: Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math. 68, 229–239 (1995)
Biler P.: The Cauchy problem and self-similar solution for a nonlinear parabolic equation. Studia Math. 114, 181–205 (1995)
Biler P.: Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)
Biler P., Brandolese L.: On the parabolic-elliptic limit of the doubly parabolic Keller–Segel system modelling chemotaxis. Studia Math. 193, 241–261 (2009)
P. Biler, L. Corrias, J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, 1–32, J. Math. Biol. doi:10.1007/s00285-010-0357-5 .
Biler P., Dolbeault J.: Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems. Ann. Henri Poincar 1(3), 461–472 (2000)
Biler P., Karch G., Laurençot P., Nadzieja T.: The \({8\pi}\) -problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Meth. Appl. Sci. 29, 1563–1583 (2006)
Blanchet A., Dolbeault J., Perthame B.: Two dimensional Keller–Segel model: optimal critical mass and qualitative properties of solutions. Electron. J. Differential Equations 44, 1–32 (2006) (electronic)
Burger M., Di Francesco M., Dolak-Struss Y.: The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38, 1288–1315 (2006)
Calvez V., Corrias L.: The parabolic-parabolic Keller-Segel model in \({\mathbb{R} ^2}\). Comm. Math. Sci. 6((2), 417–447 (2008)
Diaz J.I., Nagai T., Rakotoson J.-M.: Symmetrization techniques on unbounded domains: application to a chemotaxis system on \({\mathbb {R}^N}\). J. Differential Equations 145, 156–183 (1998)
Dolbeault J., Perthame B.: Optimal critical mass in the two dimensional Keller–Segel model in \({\mathbb {R} ^2}\). C. R. Acad. Sci. Paris, Ser. I 339, 611–616 (2004)
Duoandikoetxea J., Zuazua E.: Moments, masses de Dirac et decomposition de fonctions. C. R. Acad. Sci. Paris. Math. 315(6), 693–698 (1992)
Escobedo M., Zuazua E.: Large time behavior for convection-diffusion equation in \({\mathbb {R} ^n}\). J. Funct. Anal. 100, 119–161 (1991)
Herczak A., Olech M.: Existence and asymptotics of solutions of the Debye-Nernst-Planck system in \({\mathbb {R} ^2}\). Banach Center Publ. 86, 129–148 (2009)
Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329, 819–824 (1992)
Karch G.: Scaling in nonlinear parabolic equations. J. Math. Anal. Appl. 234, 534–558 (1999)
G. Karch, K. Suzuki, Blow-up versus global existence of solutions to aggregation equation with diffusion, (2009), 1–16, arXiv:1004.4021.
Karch G., Suzuki K.: Spikes and diffusion waves in one-dimensional model of chemotaxis. Nonlinearity 23, 1–24 (2010) arXiv:1008.0020
Kato M.: Sharp asymptotics for a parabolic system of chemotaxis in one space dimension. Diff. Integral. Eq. 22, 35–51 (2009)
Kozono H., Sugiyama Y.: Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system. J. Evol. Equ. 8, 353–378 (2008)
Lemarié-Rieusset P.G.: Recent Development in the Navier-Stokes Problem. Chapman & Hall/CRC Press, Boca Raton (2002)
Mizutani Y., Muramoto N., Yoshida K.: Self-similar radial solutions to a parabolic system modelling chemotaxis via variational method. Hiroshima Math. J. 29, 145–160 (1999)
Nagai T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)
Nagai T., Syukuinn R., Umesako M.: Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in \({\mathbb {R} ^n}\). Funk. Ekvacioj 46, 383–407 (2003)
Nagai T., Yamada T.: Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space. J. Math. Anal. Appl. 336, 704–726 (2007)
Raczyński A.: Stability property of the two-dimensional Keller-Segel model. Asymptotic Analysis 61, 35–59 (2009)
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30 , Princeton University Press, Princeton, NJ, 1970.
Wolansky G.: On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity. J. Anal. Math. 59, 251–272 (1992)
Acknowledgments
The author would like to express his gratitude to G. Karch for many stimulating discussions. It is also the pleasure to thank the referee for pertinent remarks. The preparation of this paper was partially supported by the MNiSzW grant no. N N201 418839.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Raczyński, A. Diffusion-dominated asymptotics of solution to chemotaxis model. J. Evol. Equ. 11, 509–529 (2011). https://doi.org/10.1007/s00028-011-0099-x
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DOI: https://doi.org/10.1007/s00028-011-0099-x