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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linearized stability of traveling cell solutions arising from a moving boundary problem


Authors: Y. S. Choi and Roger Lui
Journal: Proc. Amer. Math. Soc. 135 (2007), 743-753
MSC (2000): Primary 35P15; Secondary 35R35, 37L15
Published electronically: August 28, 2006
MathSciNet review: 2262870
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Abstract: In 2003, Mogilner and Verzi proposed a one-dimensional model on the crawling movement of a nematode sperm cell. Under certain conditions, the model can be reduced to a moving boundary problem for a single equation involving the length density of the bundled filaments inside the cell. It follows from the results of Choi, Lee and Lui (2004) that this simpler model possesses traveling cell solutions. In this paper, we show that the spectrum of the linear operator, obtained from linearizing the evolution equation about the traveling cell solution, consists only of eigenvalues and there exists $ \mu > 0$ such that if $ \lambda$ is a real eigenvalue, then $ \lambda \leq -\mu$. We also provide strong numerical evidence that this operator has no complex eigenvalue.


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Additional Information

Y. S. Choi
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: choi@math.uconn.edu

Roger Lui
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: rlui@wpi.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08535-2
PII: S 0002-9939(06)08535-2
Keywords: Cell motility, moving boundary problem, traveling cell, linearized operator, eigenvalues
Received by editor(s): September 3, 2004
Received by editor(s) in revised form: September 26, 2005
Published electronically: August 28, 2006
Additional Notes: The first author’s research was partially supported by NIH grant no. 5P41-RR013186-07.
The second author’s research was partially supported by NSF grant no. DMS-0456570.
Communicated by: M. Gregory Forest
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.