[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Global Existence and Blowup of Solutions of Two Species Chemotaxis Model

Journal of Applied Nonlinear Dynamics 5(4) (2016) 457--469 | DOI:10.5890/JAND.2016.12.006

V. Bhuvaneswari; K. Balachandran

Department of Mathematics, Bharathiar University, Coimbatore 641046, India

Download Full Text PDF

 

Abstract

This paper is devoted to obtain the global existence and blow up of solutions for two species chemotaxis system by the ratio of two solutions method. Our main concern is to show that the blow up properties of solutions depend only on the first eigenvalue.

References

  1. [1]  Keller, E.F. and Segel, L.A. (1970), Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26, 399-415.
  2. [2]  Lin, K., Mu, C. andWang, L. (2015), Boundedness in a two species chemotaxis system, Mathematical Methods in the Applied Sciences, 38, 5085-5096.
  3. [3]  Zhang, Z. (2006), Existence of global solution and nontrivial steady states for a system modeling chemotaxis, Abstract and Applied Analysis, Article ID 81265, 1-23.
  4. [4]  Zhang, Q. and Li, Y. (2015), Global boundedness of solutions to a two species chemotaxis system, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 66, 83-93.
  5. [5]  Horstmann, D. (2003), From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresbericht der Deutschen Mathematiker-Vereinigung, 105, 103-165.
  6. [6]  Horstmann, D. (2011), Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis and blow up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21, 231-270.
  7. [7]  Kurganov, A. and Medvid'ová, M.L. (2014), Numerical study of two species chemotaxis models, Discrete and Continuous Dynamical Systems Series B, 19, 131-152.
  8. [8]  Li, Y. and Li, Y. (2014), Finite time blow up in higher dimensional fully parabolic chemotaxis system for two species, Nonlinear Analysis, 109, 72-84.
  9. [9]  Zhang, Q. and Li, Y. (2014), Global existence and asymptotic properties of the solution to a two species chemotaxis system, Journal of Mathematical Analysis and Applications, 418, 47-63.
  10. [10]  Biler, P., Espejo, E.E. and Guerra, I. (2013), Blowup in higher dimensional two species chemotactic systems, Communications on Pure and Applied Analysis, 12, 89-98.
  11. [11]  Biler, P. and Guerra, I. (2012), Blowup and self similar solutions for two component drift- diffusion systems, Nonlinear Analysis, 75, 5186-5193.
  12. [12]  Conca, C. and Espejo, E.E. (2012), Threshold condition for global existence and blow up to a radially symmetric drift diffusion system, Applied Mathematics Letters, 25, 352-356.
  13. [13]  Conca, C., Espejo, E.E. and Vilches, K. (2011), Remarks on the blow up and global existence for a two species chemotactic Keller-Segel system in R2, European Journal of Applied Mathematics, 22, 553-580.
  14. [14]  Espejo, E.E., Stevans, A. and Velázquez, J.J.L. (2009), Simultaneous finite time blow up in a two species model for chemotaxis, Analysis, 29, 317-338.
  15. [15]  Espejo, E.E. , Stevans, A. and Velázquez,, J.J.L. (2010), A note on non-simultaneous blow up for a drift diffusion model, Differential and Integral Equations, 23, 451-462.
  16. [16]  Espejo, E.E., Vilches, K. and Conca, C. (2013), Sharp condition for blow up and global existence in a two species chemotactic Keller-Segel system in R2, European Journal of Applied Mathematics, 24, 297-313.
  17. [17]  Bhuvaneswari, V., Shangerganesh, L. and Balachandran, K. (2015), Global existence and blow up of solutions of quasilinear chemotaxis system, Mathematical Methods in the Applied Sciences, 38, 3738-3746.
  18. [18]  Hillen, T. and Painter, K.J. (2009), A user’s guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58, 183-217.
  19. [19]  Shangerganesh, L., Barani Balan, N. and Balachandran, K. (2014), Existence and uniqueness of solutions of degenerate chemotaxis system, Taiwanese Jounal of Mathematics, 18, 1605-1622.
  20. [20]  Winkler, M. (2013), Finite time blow up in the higher-dimensional parabolic-parabolic Keller-Segel system, Journal de Mathématiques Pures et Appliquées, 100, 748-767.
  21. [21]  Chen, S. (2008), Global existence and nonexistence for some degenerate and quasilinear parabolic systems, Journal of Differential Equations, 245, 1112-1136.
  22. [22]  Ladyženskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1967), Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence.
  23. [23]  Protter, M.A. and Weinberger, H.F. (1984), Maximum principles in differential equations, Springer-Verlag: New York.
  24. [24]  Li, Y., Deng, W. and Xie, C. (2002), Global existence and nonexistence for degenerate parabolic systems, Proceedings of the American Mathematical Society, 130, 3661-3670.
  25. [25]  Fuglede, B. (1999), Continuous domain dependence of the eigenvalues of the Dirichlet laplacian and related operators in hilbert space, Journal of Functional Analysis, 167, 183-200.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates