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

Harunori Monobe; Hirokazu Ninomiya

doi:10.3934/dcdsb.2014.19.789

Article Contents

2014, Volume 19, Issue 3: 789-799. Doi: 10.3934/dcdsb.2014.19.789

This issue Previous Article Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies Next Article Exponential stability of the traveling fronts for a viscous Fisher-KPP equation

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

Received: July 31, 2013

Revised: November 30, 2013

Published: April 2014

Abstract Related Papers Cited by

Abstract

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:

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

Citation:

Related Papers

Cited by

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.