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Traveling wave solutions with convex domains for a free boundary problem

  • Author Bio: E-mail address: te12001@meiji.ac.jp; E-mail address: hirokazu.ninomiya@gmail.com
The first author is partially supported by Grant-in-Aid for Research Activity Start-up (No. 20635809) from the Japan Society for the Promotion of Science. The second author is partially supported by Grant-in-Aid for Scientific Research (B) (No. 26287024) from the Japan Society for the Promotion of Science.
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  • In this paper, a free boundary problem related to cell motility is discussed. This free boundary problem consists of an interface equation for the domain evolution and a parabolic equation governing actin concentration in the domain. In [10] the existence of traveling wave solutions with disk-shaped domains were shown in a special situation where a polymerization rate is specified. In this paper, by relaxing the condition for the polymerization rate, the previous result is extended to the existence of traveling wave solutions with convex domains.

    Mathematics Subject Classification: Primary:35C07, 35R35;Secondary:92C17.

    Citation:

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  • [1] Y. S. ChoiJ. Lee and R. Lui, Traveling wave solutions for a one-dimensional crawling nematode sperm cell model, J. Math. Biol., 49 (2004), 310-328.  doi: 10.1007/s00285-003-0255-1.
    [2] Y. S. ChoiP. Groulx and R. Lui, Moving boundary problem for a one-dimensional crawling nematode sperm cell model, Nonlinear Analysis: Real World Appl., 6 (2005), 874-898.  doi: 10.1016/j.nonrwa.2004.11.005.
    [3] Y. S. Choi and R. Lui, Existence of traveling domain solutions for a two-dimensional moving boundary problem, Trans. A. M. S., 361 (2009), 4027-4044.  doi: 10.1090/S0002-9947-09-04562-0.
    [4] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998.
    [5] A. Mogilner and L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells, A quantitative analysis, Biophys. J., 83 (2002), 1237-1258. 
    [6] A. MogilnerJ. Stajic and C. W. Wolgemuth, Redundant mechanisms for stable cell locomotion revealed by minimal models, Biophys J., 101 (2011), 545-553. 
    [7] A. Mogilner and B. Rubinstein, et al., Actin-myosin viscoelastic flow in the keratocyte lamellipod, Bio. J., 97 (2009), 1853-1863. 
    [8] A. Mogilner and D. W. Verzi, A simple 1-D physical model for the crawling nematode sperm cell, J. Stat. Phys., 110 (2003), 1169-1189. 
    [9] H. Monobe, Behavior of solutions for a free boundary problem describing amoeba motion, Differential and Integral Equations, 25 (2012), 93-116. 
    [10] H. Monobe and H. Ninomiya, Multiple existence of traveling waves of a free boundary problem describing cell motility, Discrete and Continuous Dynamical Systems Series B, 19 (2014), 789-799.  doi: 10.3934/dcdsb.2014.19.789.
    [11] J. V. SmallM. Herzog and K. Anderson, Actin filament organization in the fish keratocyte lamellipodium, J. Cell Biol., 129 (1995), 1275-1286.  doi: 10.1083/jcb.129.5.1275.
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