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IMA Journal of Applied Mathematics Advance Access originally published online on February 14, 2007
IMA Journal of Applied Mathematics 2007 72(2):140-162; doi:10.1093/imamat/hxl028
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The stability and dynamics of a spike in the 1D Keller–Segel model

K. Kang1, T. Kolokolnikov2,** and M. J. Ward3

1 Department of Mathematics, Institute of Basic Science, Sungkyunkwan University, Suwon, South Korea, 2 Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada B3H 3J5, 3 Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2

** Email: t.kolokol{at}gmail.com


  Abstract

In the limit of a large mass M >> 1 and on a finite interval of length 2L, an equilibrium spike solution to the classical Keller–Segel chemotaxis model with a linear chemotactic function is constructed asymptotically. By calculating an asymptotic formula for the translational eigenvalue for M >> 1, it is shown that the equilibrium spike solution is unstable to translations of the spike profile. If in addition L >> 1, the equilibrium spike is shown to be metastable as a result of an asymptotically exponentially small eigenvalue. For M >> 1 and L >> 1, an asymptotic ordinary differential equation for the metastable spike motion is derived that shows that the spike drifts exponentially slowly towards one of the boundaries of the domain. For a certain reduced Keller–Segel model, corresponding to a domain of small length, a solution with a spike at each of the two boundaries is constructed. This solution is found to be metastable, and it is shown that there is an exponentially slow exchange of mass between the two spikes that occurs over very long timescales. For arbitrary initial conditions, energy methods are used to show the global existence of solutions. The relationship between this reduced Keller–Segel model and a Burgers-type equation modelling the upward propagation of a flame front in a finite channel is emphasized. Full numerical computations are used to confirm the asymptotic results.

Received on 20 July 2005. accepted on 18 October 2006.


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