# American Institute of Mathematical Sciences

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

 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 $T$.
Citation: Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & 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. 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. 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. 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. doi: 10.1090/S0002-9947-09-04562-0. [5] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (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. 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. 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. [9] A. Mogilner and B. Rubinstein et al, Actin-myosin viscoelastic flow in the keratocyte lamellipod,, Bio. J., 97 (2009), 1853. [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. [11] H. Monobe, Behavior of solutions for a free boundary problem describing amoeba motion,, Differential and Integral Equations, 25 (2012), 93. [12] J. V. Small, M. Herzog and K. Anderson, Actin filament organization in the fish keratocyte lamellipodium,, J. Cell Biol., 129 (1995), 1275. 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).

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##### References:
 [1] P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media,, Physica D, 94 (1996), 205. 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. 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. 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. doi: 10.1090/S0002-9947-09-04562-0. [5] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (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. 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. 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. [9] A. Mogilner and B. Rubinstein et al, Actin-myosin viscoelastic flow in the keratocyte lamellipod,, Bio. J., 97 (2009), 1853. [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. [11] H. Monobe, Behavior of solutions for a free boundary problem describing amoeba motion,, Differential and Integral Equations, 25 (2012), 93. [12] J. V. Small, M. Herzog and K. Anderson, Actin filament organization in the fish keratocyte lamellipodium,, J. Cell Biol., 129 (1995), 1275. 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).
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