# American Institute of Mathematical Sciences

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

## Chemotaxis can prevent thresholds on population density

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 and 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. [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. [3] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. [4] A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008. [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. [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. [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. [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. [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. [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. [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. [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, Providence, RI. [13] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, Preprint. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [3] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. [4] A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008. [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. [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. [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. [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. [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. [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. [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. [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, Providence, RI. [13] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system, Preprint. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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