Mathematical Challenges in the Theory of Chemotaxis

Piotr Biler

doi:10.2478/amsil-2018-0004

.blurhash-client-img { display: none !important; }

Mathematical Challenges in the Theory of Chemotaxis

Annales Mathematicae Silesianae

Volume 32 (2018): Issue 1 (September 2018)

By: Piotr Biler

Open Access

|Aug 2018

Abstract

We consider the simplest parabolic-elliptic model of chemotaxis in the whole space and in several space dimensions. Criteria either for the existence of radial global-in-time solutions or their blowup in terms of suitable Morrey spaces norms are discussed. This is an extended version of the lecture presented at the University of Silesia on January 12, 2018, commemorating Professor Andrzej Lasota-great scholar, master of fine mathematics and applications to real world.

References

[1] Andreucci D., DiBenedetto E., On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 363-441.
Search in Google Scholar Back to article
[2] Balinsky A.A., Evans W.D., Lewis R.T., The Analysis and Geometry of Hardy’s Inequality, Universitext, Springer, Cham, 2015.10.1007/978-3-319-22870-9
Search in Google Scholar Back to article
[3] Bedrossian J., Masmoudi N., Existence, uniqueness and Lipschitz dependence for Patlak-Keller-Segel and Navier-Stokes in R² with measure-valued initial data, Arch. Ration. Mech. Anal. 214 (2014), 717-801.10.1007/s00205-014-0796-z
Search in Google Scholar Back to article
[4] Bellomo N., Bellouquid A., Tao Y., Winkler M., Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), 1663-1763.10.1142/S021820251550044X
Search in Google Scholar Back to article
[5] Biler P., The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181-205.10.4064/sm-114-2-181-205
Search in Google Scholar Back to article
[6] Biler P., Existence and nonexistence of solutions for a model of gravitational interaction of particles. III, Colloq. Math. 68 (1995), 229-239.10.4064/cm-68-2-229-239
Search in Google Scholar Back to article
[7] Biler P., Growth and accretion of mass in an astrophysical model, Appl. Math. (Warsaw) 23 (1995), 179-189.10.4064/am-23-2-179-189
Search in Google Scholar Back to article
[8] Biler P., Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.
Search in Google Scholar Back to article
[9] Biler P., Radially symmetric solutions of a chemotaxis model in the plane - the supercritical case, in: Rencławowicz J., Zajaczkowski W.M. (eds.), Parabolic and Navier- Stokes Equations. Part 1, Banach Center Publications, 81, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 31-42.10.4064/bc81-0-2
Search in Google Scholar Back to article
[10] Biler P., Blowup versus global in time existence of solutions for nonlinear heat equations, Topol. Methods Nonlinear Anal. To appear. Available at arXiv:1705.03931v2.
Search in Google Scholar Back to article
[11] Biler P., Solvability for nonlinear heat equations with fractional diffusion. In preparation.
Search in Google Scholar Back to article
[12] Biler P., Brandolese L., Global existence versus blow up for some models of interacting particles, Colloq. Math. 106 (2006), 293-303.10.4064/cm106-2-9
Search in Google Scholar Back to article
[13] Biler P., Brandolese L., On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math. 193 (2009), 241-261.10.4064/sm193-3-2
Search in Google Scholar Back to article
[14] Biler P., Cieslak T., Karch G., Zienkiewicz J., Local criteria for blowup in twodimensional chemotaxis models, Discrete Contin. Dyn. Syst. 37 (2017), 1841-1856.10.3934/dcds.2017077
Search in Google Scholar Back to article
[15] Biler P., Corrias L., Dolbeault J., Large mass self-similar solutions of the parabolicparabolic Keller-Segel model of chemotaxis, J. Math. Biol. 63 (2011), 1-32.10.1007/s00285-010-0357-520730434
Open DOI Search in Google Scholar Back to article
[16] Biler P., Dolbeault J., Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000), 461-472.10.1007/s000230050003
Search in Google Scholar Back to article
[17] Biler P., Guerra I., Karch G., Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, Commun. Pure Appl. Anal. 14 (2015), 2117-2126.10.3934/cpaa.2015.14.2117
Search in Google Scholar Back to article
[18] Biler P., Hilhorst D., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. II, Colloq. Math. 67 (1994), 297-308.10.4064/cm-67-2-297-308
Search in Google Scholar Back to article
[19] Biler P., Karch G., Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ. 10 (2010), 247-262.10.1007/s00028-009-0048-0
Search in Google Scholar Back to article
[20] Biler P., Karch G., Solutions of fractional chemotaxis models. In preparation.
Search in Google Scholar Back to article
[21] Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal. 27 (2006), 133-147.
Search in Google Scholar Back to article
[22] Biler P., Karch G., Laurençot Ph., Nadzieja T., The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006), 1563-1583.10.1002/mma.743
Search in Google Scholar Back to article
[23] Biler P., Karch G., Pilarczyk D., Global radial solutions in classical Keller-Segel chemotaxis model. In preparation.
Search in Google Scholar Back to article
[24] Biler P., Karch G., Zienkiewicz J., Optimal criteria for blowup of radial and N-symmetric solutions of chemotaxis systems, Nonlinearity 28 (2015), 4369-4387.10.1088/0951-7715/28/12/4369
Search in Google Scholar Back to article
[25] Biler P., Karch G., Zienkiewicz J., Morrey spaces norms and criteria for blowup in chemotaxis models, Netw. Heterog. Media 11 (2016), 239-250.10.3934/nhm.2016.11.239
Open DOI Search in Google Scholar Back to article
[26] Biler P., Karch G., Zienkiewicz J., Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834-875.10.1016/j.aim.2018.03.036
Search in Google Scholar Back to article
[27] Biler P., Nadzieja T., Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math. 66 (1994), 319-334.10.4064/cm-66-2-319-334
Search in Google Scholar Back to article
[28] Biler P., Zienkiewicz J., Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data, Bull. Pol. Acad. Sci. Math. 63 (2015), 41-51.10.4064/ba63-1-6
Search in Google Scholar Back to article
[29] Biler P., Zienkiewicz J., Blowing up radial solutions in the minimal Keller-Segel chemotaxis model. In preparation.
Search in Google Scholar Back to article
[30] Blanchet A., Carlen E.A., Carrillo J.A., Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal. 262 (2012), 2142-2230.10.1016/j.jfa.2011.12.012
Search in Google Scholar Back to article
[31] Blanchet A., Carrillo J.A., Masmoudi N., Infinite time aggregation for the critical Patlak-Keller-Segel model in R², Comm. Pure Appl. Math. 61 (2008), 1449-1481.10.1002/cpa.20225
Search in Google Scholar Back to article
[32] Blanchet A., Dolbeault J., Perthame B., Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006, no. 44, 32 pp.
Search in Google Scholar Back to article
[33] Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A., Venkataramani S.C., Diffusion, attraction and collapse, Nonlinearity 12 (1999), 1071-1098.10.1088/0951-7715/12/4/320
Open DOI Search in Google Scholar Back to article
[34] Chandrasekhar S., Principles of Stellar Dynamics, University of Chicago Press, Chicago, 1942.
Search in Google Scholar Back to article
[35] Chavanis P.H., Sommeria J., Robert R., Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophys. Journal 471 (1996), 385-399.10.1086/177977
Search in Google Scholar Back to article
[36] Corrias L., Perthame B., Zaag H., Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004), 1-28.10.1007/s00032-003-0026-x
Search in Google Scholar Back to article
[37] Debye P., Hückel E., Zur Theorie der Electrolyte. II, Phys. Zft. 24 (1923), 305-325.
Search in Google Scholar Back to article
[38] Fujita H., On the blowing up of solutions of the Cauchy problem for ut = Δu+u^{1+ α}, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109-124.
Search in Google Scholar Back to article
[39] Giga Y., Miyakawa T., Navier-Stokes flow in R³ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577-618.10.1080/03605308908820621
Open DOI Search in Google Scholar Back to article
[40] Giga Y., Mizoguchi N., Senba T., Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Arch. Ration. Mech. Anal. 201 (2011), 549-573.10.1007/s00205-010-0394-7
Search in Google Scholar Back to article
[41] Herrero M.A., Velázquez J.J.L., Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583-623.10.1007/BF01445268
Search in Google Scholar Back to article
[42] Herrero M.A., Velázquez J.J.L., Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177-194.10.1007/s0028500500499053436
Open DOI Search in Google Scholar Back to article
[43] Hillen T., Painter K.J., A users guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183-217.10.1007/s00285-008-0201-3
Open DOI Search in Google Scholar Back to article
[44] Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), 103-165.
Search in Google Scholar Back to article
[45] Horstmann D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), 51-69.
Search in Google Scholar Back to article
[46] Iwabuchi T., Global well-posedness for Keller-Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011), 930-948.10.1016/j.jmaa.2011.02.010
Search in Google Scholar Back to article
[47] Jäger W., Luckhaus S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824.10.1090/S0002-9947-1992-1046835-6
Search in Google Scholar Back to article
[48] Karch G., Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999), 534-558.10.1006/jmaa.1999.6370
Search in Google Scholar Back to article
[49] Kavallaris N.I., Souplet Ph., Grow-up rate and refined asymptotics for a twodimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal. 40 (2008/09), 1852-1881.10.1137/080722229
Search in Google Scholar Back to article
[50] Keller E.F., Segel L.A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.10.1016/0022-5193(70)90092-5
Open DOI Search in Google Scholar Back to article
[51] Kozono H., Sugiyama Y., The Keller-Segel system of parabolic-parabolic type with initial data in weak L^n/2(Rⁿ) and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1467-1500.10.1512/iumj.2008.57.3316
Search in Google Scholar Back to article
[52] Kurokiba M., Ogawa T., Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations 16 (2003), 427-452.10.57262/die/1356060652
Search in Google Scholar Back to article
[53] Lemarié-Rieusset P.-G., Small data in an optimal Banach space for the parabolicparabolic and parabolic-elliptic Keller-Segel equations in the whole space, Adv. Differential Equations 18 (2013), 1189-1208.10.57262/ade/1378327383
Search in Google Scholar Back to article
[54] Mizoguchi N., Senba T., Type-II blowup of solutions to an elliptic-parabolic system, Adv. Math. Sci. Appl. 17 (2007), 505-545.
Search in Google Scholar Back to article
[55] Mizoguchi N., Senba T., A sufficient condition for type I blowup in a parabolic-elliptic system, J. Differential Equations 250 (2011), 182-203.10.1016/j.jde.2010.10.016
Search in Google Scholar Back to article
[56] Nagai T., Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001), 37-55.10.1155/S1025583401000042
Search in Google Scholar Back to article
[57] Naito Y., Senba T., Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains, Discrete Contin. Dyn. Syst. 32 (2012), 3691-3713.10.3934/dcds.2012.32.3691
Search in Google Scholar Back to article
[58] Naito Y., Senba T., Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space, Commun. Pure Appl. Anal. 12 (2013), 1861-1880.10.3934/cpaa.2013.12.1861
Open DOI Search in Google Scholar Back to article
[59] Pilarczyk D., Asymptotic stability of singular solution to nonlinear heat equation, Discrete Contin. Dyn. Syst. 25 (2009), 991-1001.10.3934/dcds.2009.25.991
Search in Google Scholar Back to article
[60] Pilarczyk D., Self-similar asymptotics of solutions to heat equation with inverse square potential, J. Evol. Equ. 13 (2013), 69-87.10.1007/s00028-012-0169-8
Open DOI Search in Google Scholar Back to article
[61] Quittner P., Souplet Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.
Search in Google Scholar Back to article
[62] Senba T., Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domains, Funkcial. Ekvac. 48 (2005), 247-271.10.1619/fesi.48.247
Search in Google Scholar Back to article
[63] Souplet Ph., Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in Rⁿ, J. Funct. Anal. 272 (2017), 2005-2037.10.1016/j.jfa.2016.09.002
Search in Google Scholar Back to article
[64] Suzuki T., Free Energy and Self-Interacting Particles, Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser, Boston, 2005.10.1007/0-8176-4436-9
Search in Google Scholar Back to article
[65] Taylor M.E., Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407-1456.10.1080/03605309208820892
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2018-0004 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

Journal RSS Feed

Language: English

Page range: 43 - 63

Submitted on: Feb 22, 2018

Accepted on: Mar 21, 2018

Published on: Aug 24, 2018

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

chemotaxis,

blowup of solutions,

global existence of solutions

Related subjects:

Mathematics,

General mathematics

© 2018 Piotr Biler, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 32 (2018): Issue 1 (September 2018)