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

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