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

Chemotaxis can prevent thresholds on population density

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.
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.  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.  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, (1998).   Google Scholar

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A. Friedman, Partial Differential Equations,, Dover Books on Mathematics Series, (2008).   Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   Google Scholar

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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.   Google Scholar

[7]

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

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

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

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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.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

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

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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.  doi: 10.1155/S1025583401000042.  Google Scholar

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T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   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.  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.  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.  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.  doi: 10.1137/13094058X.  Google Scholar

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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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (1998).   Google Scholar

[4]

A. Friedman, Partial Differential Equations,, Dover Books on Mathematics Series, (2008).   Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001).   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.   Google Scholar

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  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.   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.   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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1007/s00332-014-9205-x.  Google Scholar

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