February  2017, 37(2): 905-914. doi: 10.3934/dcds.2017037

Traveling wave solutions with convex domains for a free boundary problem

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, Japan

Received  March 2015 Revised  September 2015 Published  November 2016

Fund Project: 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

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.

Citation: Harunori Monobe, Hirokazu Ninomiya. Traveling wave solutions with convex domains for a free boundary problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 905-914. doi: 10.3934/dcds.2017037
References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998.Google Scholar

[5]

A. Mogilner and L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells, A quantitative analysis, Biophys. J., 83 (2002), 1237-1258. Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

show all references

References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998.Google Scholar

[5]

A. Mogilner and L. Edelstein-Keshet, Regulation of actin dynamics in rapidly moving cells, A quantitative analysis, Biophys. J., 83 (2002), 1237-1258. Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

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