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July  2015, 20(5): 1499-1527. doi: 10.3934/dcdsb.2015.20.1499

Chemotaxis can prevent thresholds on population density

Johannes Lankeit 1,

1. 

Institut für Mathematik, Universitat Paderborn, 33098 Paderborn, Germany

Received  March 2014 Revised  January 2015 Published  May 2015

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to \begin{align*} u_t = -\nabla \cdot (u \nabla v)+\kappa u-\mu u^2\\ 0 = \Delta v - v + u      \\ \partial_v v|\partial \Omega = \partial_v u|\partial \Omega = 0 ,           u(0,\cdot) = u_0, \end{align*} in balls in $\mathbb{R}^n$. They exist globally in time for $\mu\ge 1$ and, for a certain class of initial data, undergo finite-time blow-up if $\mu<1$.
    We then use this blow-up result to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler [26] to the higher dimensional (radially symmetric) case.
Keywords: hyperbolic-elliptic system., Chemotaxis, blow-up, logistic source.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B44, 35A01, 35A02, 35Q92, 92C1.
Citation: Johannes Lankeit. Chemotaxis can prevent thresholds on population density. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1499-1527. doi: 10.3934/dcdsb.2015.20.1499
References:
[1]

V. Andasari, A. Gerisch, G. Lolas, A. P. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1.  Google Scholar

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A. Aotani, M. Mimura and T. Mollee, A model aided understanding of spot pattern formation in chemotactic E. coli colonies, Jpn. J. Ind. Appl. Math., 27 (2010), 5-22. doi: 10.1007/s13160-010-0011-z.  Google Scholar

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L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[4]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683 (1998), URL http://www.numdam.org/item?id=ASNSP_1997_4_24_4_633_0.  Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[12]

O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, translations of mathematical monographs vol. 23, 1991,, American Mathematical Society, ().   Google Scholar

[13]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system,, Preprint., ().   Google Scholar

[14]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[15]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433, URL http://www.math.kobe-u.ac.jp/~fe/xml/mr1610709.xml.  Google Scholar

[16]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[18]

O. Stancevic, C. N. Angstmann, J. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75 (2013), 774-795. doi: 10.1007/s11538-013-9834-5.  Google Scholar

[19]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.  Google Scholar

[20]

Z. Szymańska, C. M. Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425.  Google Scholar

[21]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar

[22]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[23]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.  Google Scholar

[24]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[25]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[26]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x.  Google Scholar

show all references

References:
[1]

V. Andasari, A. Gerisch, G. Lolas, A. P. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1.  Google Scholar

[2]

A. Aotani, M. Mimura and T. Mollee, A model aided understanding of spot pattern formation in chemotactic E. coli colonies, Jpn. J. Ind. Appl. Math., 27 (2010), 5-22. doi: 10.1007/s13160-010-0011-z.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[4]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[6]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633-683 (1998), URL http://www.numdam.org/item?id=ASNSP_1997_4_24_4_633_0.  Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[12]

O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, translations of mathematical monographs vol. 23, 1991,, American Mathematical Society, ().   Google Scholar

[13]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system,, Preprint., ().   Google Scholar

[14]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[15]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433, URL http://www.math.kobe-u.ac.jp/~fe/xml/mr1610709.xml.  Google Scholar

[16]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. TMA, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[18]

O. Stancevic, C. N. Angstmann, J. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bull. Math. Biol., 75 (2013), 774-795. doi: 10.1007/s11538-013-9834-5.  Google Scholar

[19]

C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007. doi: 10.1137/13094058X.  Google Scholar

[20]

Z. Szymańska, C. M. Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425.  Google Scholar

[21]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar

[22]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[23]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.  Google Scholar

[24]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[25]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[26]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, Journal of Nonlinear Science, 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x.  Google Scholar

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