Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T17:48:40.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in 2

Published online by Cambridge University Press:  07 December 2012

ELIO ESPEJO ,
KARINA VILCHES  and
CARLOS CONCA
ELIO ESPEJO
Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel email: eespejo@techunix.technion.ac.il
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: kvilches@dim.uchile.cl
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: kvilches@dim.uchile.cl Institute for Cell Dynamics and Biotechnology: A Centre for Systems Biology, University of Chile, Santiago, Chile email: cconca@dim.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

Keywords

chemotaxismulticomponent Keller–Segel modelsharp conditions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 2 , April 2013 , pp. 297 - 313
Copyright
Copyright © Cambridge University Press 2012

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.)

References

[1]Attouch, H., Buttazzo, G. & Michaille, G. (2006) Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM Series on Optimization, SIAM Philadelphia.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]Blanchet, A., Carrillo, J. & Masmoudi, N. (2008) Infinite Time Aggregation for the Critical Patlak–Keller–Segel Model in 2. Commun. Pure Appl. Math. LXI, 1449–1481.Google Scholar
[4]Blanchet, A., Dolbeault, J. & Perthame, B. (2006) Two-dimensional Keller–Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44, 1–32.Google Scholar
[5]Childress, S. & Percus, J. K. (1984) Chemotactic Collapse in Two Dimensions, Lecture Notes in Biomathematics, Vol. 56, Springer, Berlim, Germany, pp. 217–237.Google Scholar
[6]Conca, C., Espejo, E. & Vilches, K. (2011) Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in 2. Eur. J. Appl. Math. 22, 553–580 doi:10.1017/S0956792511000258.Google Scholar
[7]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.Google Scholar
[8]Fonseca, I. & Leoni, G. (2007) Modern Methods in the Calculus of Variations: Lp Spaces, Springer Monographs in Mathematics, Springer, Berlin, Germany.Google Scholar
[9]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
[10]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
[11]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.Google Scholar
[12]Keller, E. F. & Segel, L. A. (1971) Traveling bands of chemotactic bacteria. J. Theor. Biol. 30, 235–248.Google Scholar
[13]Nagai, T. (1995) Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601.Google Scholar
[14]Shafrir, I., & Wolansky, G. (2005) Moser-Trudinger and logarithmic HLS inequalities for systems. J. Eur. Math. Soc. 4, 413–448.Google Scholar
[15]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).Google Scholar
[16]Wolansky, G. (2002) Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13, 641–661.Google Scholar