Hostname: page-component-7dd5485656-s6l46 Total loading time: 0 Render date: 2025-10-31T11:44:56.374Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in 2

Published online by Cambridge University Press:  20 July 2011

CARLOS CONCA ,
ELIO ESPEJO  and
KARINA VILCHES
CARLOS CONCA
Affiliation:
Departamento de Ingeniería Matemática (DIM) and Centro de Modelamiento Matemático (CMM), Universidad de Chile, UMI CNRS 2807, Casilla 170-3, Correo 3, Santiago, Chile email: cconca@dim.uchile.cl, kvilches@dim.uchile.cl Institute for Cell Dynamics and Biotechnology: A Centre for Systems Biology, University of Chile, Santiago, Chile
ELIO ESPEJO
Affiliation:
Millennium Institute for Cell Dynamics and Biotechnology, University of Chile, Santiago, Chile email: eespejo@ing.uchile.cl
KARINA VILCHES
Affiliation:
Departamento de Ingeniería Matemática (DIM) and Centro de Modelamiento Matemático (CMM), Universidad de Chile, UMI CNRS 2807, Casilla 170-3, Correo 3, Santiago, Chile email: cconca@dim.uchile.cl, kvilches@dim.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

For the Keller–Segel model, it was conjectured by Childress and Percus (1984, Chemotactic collapse in two dimensions. In Lecture Notes in Biomath. Vol. 55, Springer, Berlin-Heidelberg-New York, 1984, pp. 61–66) that in a two-dimensional domain there exists a critical number C such that if the initial mass is strictly less than C, then the solution exists globally in time and if it is strictly larger than C blowup happens. For different versions of the Keller–Segel model, the conjecture has essentially been proved. The case of several chemotactic species introduces an additional question: What is the analogue for the critical mass obtained for the single species system? In this paper, we investigate for a two-species model for chemotaxis in 2 the conditions on the initial data, which determine blowup or global existence in time. Specifically, we find a curve in the plane of masses such that outside of it there is blowup and inside of it global existence in time is proved when the initial masses satisfy a threshold condition. Optimality of this condition is discussed through an analysis in the radial case. Finally, we show in the case of blowup for general data how it is possible to obtain a balance between entropies and prove what species should aggregates first.

Keywords

ChemotaxisMulti-component Keller–Segel modelBlowup of solutions

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 6 , December 2011 , pp. 553 - 580
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Blanchet, A., Dolbeault, J. & Perthame, B. (2006) Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Diff. Eqns. 44, 1–32.Google Scholar
[2]Biler, P. (1998) Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743.Google Scholar
[3]Carlen, E. & Loss, M. (1992) Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on Sn. Geom. Funct. Anal. 2, 90–104.CrossRefGoogle Scholar
[4]Childress, S. & Percus, J. K. (1984) Chemotactic collapse in two dimensions. In: Lecture Notes in Biomathematics, Vol. 55, Springer, Berlin-Heidelberg-New York, pp. 61–66.Google Scholar
[5]Dolbeault, J. & Perthame, B. (2004) Optimal critical mass in the two-dimensional Keller–Segel model in 2. C. R. Math. Acad. Sci. Paris Serie I 339, 611–616.CrossRefGoogle Scholar
[6]Espejo, E., Stevens, A. & Velázquez, J. J. L. (2009) Simultaneous finite time blow-up in a two-species model for chemotaxis. Analysis 29, 317–338.CrossRefGoogle Scholar
[7]Fasano, A., Mancini, A. & Primicerio, M. (2004) Equilibrium of two populations subject to chemotaxis. Math. Models Meth. App. Sci. 14, 503–533.CrossRefGoogle Scholar
[8]Gajewski, H. & Zacharias, K. (1998) Global behaviour of a reaction–diffusion system modelling chemotaxis. Math. Nachr. 195, 77–114.CrossRefGoogle Scholar
[9]Jäger, W. & Luckhaus, S. (1992) On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329, 819–824.CrossRefGoogle Scholar
[10]Horstmann, D., (2011) Generalizing Keller–Segel: Lyapunov functionals, steady state analysis and blow-up results for multi-species chemotaxis in the presence of attraction and repulsion between competitive interacting species. J. Nonlinear Sci., 21, 231–270.CrossRefGoogle Scholar
[11]Horstmann, D. & Lucia, M. (2009) Nonlocal elliptic boundary value problems related to chemotactic movement of mobile species. RIMS Kôkyûroku Bessatsu B15, 39–72.Google Scholar
[12]Horstmann, D. (2003) From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I. Jahresber. Dutsch. Math. Ver. 105, 103–165.Google Scholar
[13]Horstmann, D. (2004) From 1970 until present: The Keller–Segel model in chemotaxis and its consequences II. Jahresber. Dutsch. Math. Ver. 106, 51–69.Google Scholar
[14]Keller, E. F. & Segel, L. A. (1971) Traveling bands of chemotactic bacteria. J. Theor. Biol. 30, 235–248.CrossRefGoogle ScholarPubMed
[15]Kurokiba, M. & Ogawa, T. (2003) Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type. Diff. Integral Eqns. 4, 427–452.Google Scholar
[16]Kurokiba, M., Nagai, T. & Ogawa, T. (2006) The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Comm. Pure App. Anal. 5, 97–106.CrossRefGoogle Scholar
[17]Nagai, T. (1995) Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601.Google Scholar
[18]Nagai, T., Senba, T. & Yoshida, K. (1997) Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40, 411–433.Google Scholar
[19]Shafrir, I. & Wolansky, G. (2005) Moser–Trudinger and logarithmic HLS inequalities for systems. J. Eur. Math. Soc. 7, 413–448.CrossRefGoogle Scholar
[20]Velázquez, J. J. L. (2004) Point dynamics in a singular limit of the Keller–Segel model II. Formation of the concentration regions. SIAM J. Appl. Math. 64, 1224–1248 (electronic).CrossRefGoogle Scholar
[21]Wolansky, G. (2002) Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13, 641–661.CrossRefGoogle Scholar