Advanced Search
Article Contents
Article Contents

Chemotaxis can prevent thresholds on population density

Abstract / Introduction Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35K55; Secondary: 35B44, 35A01, 35A02, 35Q92, 92C17.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


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


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


    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.


    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.


    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.


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


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


    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.


    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.


    O. Ladyzhenskaya, V. Solonnikov and N. UraltsevaLinear and Quasi-Linear Equations of Parabolic Type, translations of mathematical monographs vol. 23, 1991, American Mathematical Society, Providence, RI.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(217) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint