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May  2014, 19(3): 789-799. doi: 10.3934/dcdsb.2014.19.789

Multiple existence of traveling waves of a free boundary problem describing cell motility

Harunori Monobe 1, and  Hirokazu Ninomiya 2,

1. 

Meiji Institute of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan
2. 

School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525

Received  August 2013 Revised  December 2013 Published  February 2014

In this paper we consider a free boundary problem describing cell motility, which is a simple model of Umeda (see [11]). This model includes a non-local term and the interface equation with curvature. We prove that there exist at least two traveling waves of the model. First, we rewrite the problem into a fixed-point problem for a continuous map $T$ and then show that there exist at least two fixed points for the map $T$.
Keywords: free boundary problems, cell crawling., Traveling waves.
Mathematics Subject Classification: Primary: 35C07, 35R35; Secondary: 92C1.
Citation: Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789
References:
[1]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D, 94 (1996), 205-220. doi: 10.1016/0167-2789(96)00042-5.

[2]

Y. S. Choi, J. 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.

[3]

Y. S. Choi, P. Groulxb 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.

[4]

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.

[5]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. doi: 10.1007/978-3-642-61798-0.

[6]

J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagation wave segments in wave front interaction model, Physica D, 239 (2010), 230-239. doi: 10.1016/j.physd.2009.11.001.

[7]

A. Mogilner and L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells, A quantitative analysis. Biophys. J., 83 (2002), 1237-1258. doi: 10.1016/S0006-3495(02)73897-6.

[8]

A. Mogilner, J. Stajic and C. W. Wolgemuth, Redundant mechanisms for stable cell locomotion revealed by minimal models, Biophys J., 101 (2011), 545-553.

[9]

A. Mogilner and B. Rubinstein et al, Actin-myosin viscoelastic flow in the keratocyte lamellipod, Bio. J., 97 (2009), 1853-1863.

[10]

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.

[11]

H. Monobe, Behavior of solutions for a free boundary problem describing amoeba motion, Differential and Integral Equations, 25 (2012), 93-116.

[12]

J. V. Small, M. 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.

[13]

V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagation of chemical waves segments, Phys. Rev. Lett., 94 (2005), 068302.

show all references

References:
[1]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Physica D, 94 (1996), 205-220. doi: 10.1016/0167-2789(96)00042-5.

[2]

Y. S. Choi, J. 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.

[3]

Y. S. Choi, P. Groulxb 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.

[4]

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.

[5]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. doi: 10.1007/978-3-642-61798-0.

[6]

J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagation wave segments in wave front interaction model, Physica D, 239 (2010), 230-239. doi: 10.1016/j.physd.2009.11.001.

[7]

A. Mogilner and L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells, A quantitative analysis. Biophys. J., 83 (2002), 1237-1258. doi: 10.1016/S0006-3495(02)73897-6.

[8]

A. Mogilner, J. Stajic and C. W. Wolgemuth, Redundant mechanisms for stable cell locomotion revealed by minimal models, Biophys J., 101 (2011), 545-553.

[9]

A. Mogilner and B. Rubinstein et al, Actin-myosin viscoelastic flow in the keratocyte lamellipod, Bio. J., 97 (2009), 1853-1863.

[10]

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.

[11]

H. Monobe, Behavior of solutions for a free boundary problem describing amoeba motion, Differential and Integral Equations, 25 (2012), 93-116.

[12]

J. V. Small, M. 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.

[13]

V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagation of chemical waves segments, Phys. Rev. Lett., 94 (2005), 068302.

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