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"Large" strange attractors in the unfolding of a heteroclinic attractor
Self polarization and traveling wave in a model for cell crawling migration
1. | Map5, UMR 8145 CNRS, Université de Paris, France |
2. | Department of Mathematics, University of Maryland, College Park MD 20742 USA |
3. | LaMME, UMR 8071 CNRS, Université Évry Val d'Essonne, France |
In this paper, we prove the existence of traveling wave solutions for an incompressible Darcy's free boundary problem recently introduced in [
References:
[1] |
L. Berlyand, J. Fuhrmann and V. Rybalko,
Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility, Commun. Math. Sci., 16 (2018), 735-762.
doi: 10.4310/CMS.2018.v16.n3.a6. |
[2] |
L. Berlyand, M. Potomkin and V. Rybalko,
Phase-field model of cell motility: Traveling waves and sharp interface limit, C. R. Math. Acad. Sci. Paris, 354 (2016), 986-992.
doi: 10.1016/j.crma.2016.09.001. |
[3] |
L. Berlyand, M. Potomkin and V. Rybalko,
Sharp interface limit in a phase field model of cell motility, Netw. Heterog. Media, 12 (2017), 551-590.
doi: 10.3934/nhm.2017023. |
[4] |
L. Berlyand and V. Rybalko, Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility, Submitted, available on arXiv.org, 2019. |
[5] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[6] |
A. Cucchi, A. Mellet and N. Meunier,
A Cahn-Hilliard model for cell motility, SIAM J. Math. Anal., 52 (2020), 3843-3880.
doi: 10.1137/19M1267969. |
[7] |
C. Etchegaray, N. Meunier and R. Voituriez,
Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration, SIAM J. Appl. Math., 77 (2017), 2040-2065.
doi: 10.1137/16M1088715. |
[8] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
doi: 10.3934/dcdsb.2004.4.147. |
[9] |
A. Friedman and B. Hu,
Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.
doi: 10.1016/j.jde.2005.09.008. |
[10] |
A. Friedman and F. Reitich,
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, rans. Amer. Math. Soc., 353 (2001), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
[11] |
M. Günther and G. Prokert,
On travelling wave solutions for a moving boundary problem of Hele-Shaw type, IMA J. Appl. Math., 74 (2009), 107-127.
doi: 10.1093/imamat/hxn029. |
[12] |
K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot,
Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480.
|
[13] |
I. Lavi, N. Meunier, R. Voituriez and J. Casademunt,
Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102.
|
[14] |
H. Levine, D. Shao and J. W. Rappel,
Coupling actin flow, adhesion, and morphology in a computational cell motility model, Proc Nat Acad Sci, 109 (2015), 6851-6856.
|
[15] |
M. S. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang,
On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25.
doi: 10.1016/j.physd.2015.10.008. |
[16] |
M. S. Mizuhara and P. Zhang,
Uniqueness and traveling waves in a cell motility model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2811-2835.
doi: 10.3934/dcdsb.2018315. |
[17] |
F. Otto,
Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.
doi: 10.1007/s002050050073. |
[18] |
T. Putelat, P. Recho and L. Truskinovsky,
Contraction-driven cell motility, Phys Rev Lett, 111 (2013), 108102.
|
[19] |
F. Ziebert and I. Aronson, Computational Approaches to Substrate-based Cell Motility, Nature Phys. Journal, 2016. |
show all references
References:
[1] |
L. Berlyand, J. Fuhrmann and V. Rybalko,
Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility, Commun. Math. Sci., 16 (2018), 735-762.
doi: 10.4310/CMS.2018.v16.n3.a6. |
[2] |
L. Berlyand, M. Potomkin and V. Rybalko,
Phase-field model of cell motility: Traveling waves and sharp interface limit, C. R. Math. Acad. Sci. Paris, 354 (2016), 986-992.
doi: 10.1016/j.crma.2016.09.001. |
[3] |
L. Berlyand, M. Potomkin and V. Rybalko,
Sharp interface limit in a phase field model of cell motility, Netw. Heterog. Media, 12 (2017), 551-590.
doi: 10.3934/nhm.2017023. |
[4] |
L. Berlyand and V. Rybalko, Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility, Submitted, available on arXiv.org, 2019. |
[5] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[6] |
A. Cucchi, A. Mellet and N. Meunier,
A Cahn-Hilliard model for cell motility, SIAM J. Math. Anal., 52 (2020), 3843-3880.
doi: 10.1137/19M1267969. |
[7] |
C. Etchegaray, N. Meunier and R. Voituriez,
Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration, SIAM J. Appl. Math., 77 (2017), 2040-2065.
doi: 10.1137/16M1088715. |
[8] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
doi: 10.3934/dcdsb.2004.4.147. |
[9] |
A. Friedman and B. Hu,
Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.
doi: 10.1016/j.jde.2005.09.008. |
[10] |
A. Friedman and F. Reitich,
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, rans. Amer. Math. Soc., 353 (2001), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
[11] |
M. Günther and G. Prokert,
On travelling wave solutions for a moving boundary problem of Hele-Shaw type, IMA J. Appl. Math., 74 (2009), 107-127.
doi: 10.1093/imamat/hxn029. |
[12] |
K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot,
Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480.
|
[13] |
I. Lavi, N. Meunier, R. Voituriez and J. Casademunt,
Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102.
|
[14] |
H. Levine, D. Shao and J. W. Rappel,
Coupling actin flow, adhesion, and morphology in a computational cell motility model, Proc Nat Acad Sci, 109 (2015), 6851-6856.
|
[15] |
M. S. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang,
On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25.
doi: 10.1016/j.physd.2015.10.008. |
[16] |
M. S. Mizuhara and P. Zhang,
Uniqueness and traveling waves in a cell motility model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2811-2835.
doi: 10.3934/dcdsb.2018315. |
[17] |
F. Otto,
Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.
doi: 10.1007/s002050050073. |
[18] |
T. Putelat, P. Recho and L. Truskinovsky,
Contraction-driven cell motility, Phys Rev Lett, 111 (2013), 108102.
|
[19] |
F. Ziebert and I. Aronson, Computational Approaches to Substrate-based Cell Motility, Nature Phys. Journal, 2016. |

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