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

Choi, Y.; Lui, Roger

doi:10.1090/S0002-9939-06-08535-2

Linearized stability of traveling cell solutions arising from a moving boundary problem
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by Y. S. Choi and Roger Lui PDF

Proc. Amer. Math. Soc. 135 (2007), 743-753 Request permission

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.

References

Y. S. Choi, Juliet Lee, and Roger Lui, Traveling wave solutions for a one-dimensional crawling nematode sperm cell model, J. Math. Biol. 49 (2004), no. 3, 310–328. MR 2102761, DOI 10.1007/s00285-003-0255-1
Y. S. Choi, Patrick Groulx, and Roger Lui, Moving boundary problem for a one-dimensional crawling nematode sperm cell model, Nonlinear Anal. Real World Appl. 6 (2005), no. 5, 874–898. MR 2165218, DOI 10.1016/j.nonrwa.2004.11.005
Mogilner, A. and D. W. Verzi (2003). A Simple 1-D Physical Model for the Crawling Nematode Sperm Cell. J. Stat. Phys. 110, 1169-1189.