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Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis. (English) Zbl 1230.92011

Summary: In two space dimensions, the parabolic-parabolic Keller-Segel system shares many properties with the parabolic-elliptic Keller-Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold \(M_{c}\). However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above \(M_{c}\) always blow up. We study forward self-similar solutions of the parabolic-parabolic Keller-Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above \(M_{c}\), which is forbidden in the parabolic-elliptic case.

MSC:

92C17 Cell movement (chemotaxis, etc.)
35K40 Second-order parabolic systems
35J60 Nonlinear elliptic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:

[1] Biler P (1998) Local and global solvability of some parabolic systems modelling chemotaxis. Adv Math Sci Appl 8(2): 715–743 · Zbl 0913.35021
[2] Biler P (2006) A note on the paper of Y. Naito: ”Asymptotically self-similar solutions for the parabolic system modelling chemotaxis”. In: Self-similar solutions of nonlinear PDE, vol 74. Banach Center Publications, Polish Academy of Sciences, Warsaw, pp 33–40 · Zbl 1116.35054
[3] Biler P, Karch G, Laurençot P, Nadzieja T (2006) The 8 {\(\pi\)}-problem for radially symmetric solutions of a chemotaxis model in the plane. Math Methods Appl Sci 29(13): 1563–1583 · Zbl 1105.35131 · doi:10.1002/mma.743
[4] Blanchet A, Carrillo JA, Masmoudi N (2008) Infinite time aggregation for the critical Patlak-Keller-Segel model in $${\(\backslash\)mathbb{R}\^2}$$ . Comm Pure Appl Math 61(10): 1449–1481 · Zbl 1155.35100 · doi:10.1002/cpa.20225
[5] Blanchet A, Dolbeault J, Perthame B (2006) Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Electron J Differ Equ 44: 32 · Zbl 1112.35023
[6] Brezis H, Merle F (1991) Uniform estimates and blow-up behavior for solutions of u = V(x) e u in two dimensions. Comm Partial Differ Equ 16(8-9): 1223–1253 · Zbl 0746.35006 · doi:10.1080/03605309108820797
[7] Calvez V, Corrias L (2008) The parabolic–parabolic Keller-Segel model in $${\(\backslash\)mathbb{R}\^2}$$ . Commun Math Sci 6(2): 417–447 · Zbl 1149.35360 · doi:10.4310/CMS.2008.v6.n2.a8
[8] Cieślak T, Laurençot P (2009) Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system. CR Math Acad Sci Paris 347(5-6): 237–242 · Zbl 1175.35010 · doi:10.1016/j.crma.2009.01.016
[9] Dolbeault J, Perthame B (2004) Optimal critical mass in the two-dimensional Keller-Segel model in $${\(\backslash\)mathbb{R}\^2}$$ . CR Math Acad Sci Paris 339(9): 611–616 · Zbl 1056.35076 · doi:10.1016/j.crma.2004.08.011
[10] Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58(1-2): 183–217 · Zbl 1161.92003 · doi:10.1007/s00285-008-0201-3
[11] Horstmann D (2002) On the existence of radially symmetric blow-up solutions for the Keller-Segel model. J Math Biol 44(5): 463–478 · Zbl 1053.35064 · doi:10.1007/s002850100134
[12] Horstmann D (2003) From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber Deutsch Math-Verein 105(3): 103–165 · Zbl 1071.35001
[13] Mizutani Y, Muramoto N, Yoshida K (1999) Self-similar radial solutions to a parabolic system modelling chemotaxis via variational method. Hiroshima Math J 29:145–160 · Zbl 0938.35187
[14] Muramoto N, Naito Y, Yoshida K (2000) Existence of self-similar solutions to a parabolic system modelling chemotaxis. Japan J Indust Appl Math 17: 427–451 · Zbl 1306.92010 · doi:10.1007/BF03167376
[15] Naito Y (2006) Asymptotically self-similar solutions for the parabolic system modelling chemotaxis. In: Self-similar solutions of nonlinear PDE, vol 74. Banach Center Publications, Polish Academy of Sciences, Warsaw, pp 149–160 · Zbl 1115.35059
[16] Naito Y, Suzuki T, Yoshida K (2002) Self-similar solutions to a parabolic system modeling chemotaxis. J Differ Equ 184(2): 386–421 · Zbl 1016.35037 · doi:10.1006/jdeq.2001.4146
[17] Raczyński A (2009) Stability property of the two-dimensional Keller–Segel model. Asymptot Anal 61: 35–59 · Zbl 1184.35153
[18] Tindall MJ, Maini PK, Porter SL, Armitage JP (2008a) Overview of mathematical approaches used to model bacterial chemotaxis. II. Bacterial populations. Bull Math Biol 70(6): 1570–1607 · Zbl 1209.92006 · doi:10.1007/s11538-008-9322-5
[19] Tindall MJ, Porter SL, Maini PK, Gaglia G, Armitage JP (2008b) Overview of mathematical approaches used to model bacterial chemotaxis. I. The single cell. Bull Math Biol 70(6): 1525–1569 · Zbl 1166.92008 · doi:10.1007/s11538-008-9321-6
[20] Yoshida K (2001) Self-similar solutions of chemotactic system. Nonlinear Anal 47: 813–824 · Zbl 1042.35550 · doi:10.1016/S0362-546X(01)00225-5
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