### Abstract

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic be- havior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlin- ear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parame- ter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

Original language | English (US) |
---|---|

Pages (from-to) | 551-590 |

Number of pages | 40 |

Journal | Networks and Heterogeneous Media |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Engineering(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Networks and Heterogeneous Media*,

*12*(4), 551-590. https://doi.org/10.3934/nhm.2017023

}

*Networks and Heterogeneous Media*, vol. 12, no. 4, pp. 551-590. https://doi.org/10.3934/nhm.2017023

**Sharp interface limit in a phase field model of cell motility.** / Berlyand, Leonid V.; Potomkin, Mykhailo; Rybalko, Volodymyr.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sharp interface limit in a phase field model of cell motility

AU - Berlyand, Leonid V.

AU - Potomkin, Mykhailo

AU - Rybalko, Volodymyr

PY - 2017/12/1

Y1 - 2017/12/1

N2 - We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic be- havior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlin- ear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parame- ter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

AB - We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic be- havior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlin- ear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parame- ter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

UR - http://www.scopus.com/inward/record.url?scp=85037147213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037147213&partnerID=8YFLogxK

U2 - 10.3934/nhm.2017023

DO - 10.3934/nhm.2017023

M3 - Article

AN - SCOPUS:85037147213

VL - 12

SP - 551

EP - 590

JO - Networks and Heterogeneous Media

JF - Networks and Heterogeneous Media

SN - 1556-1801

IS - 4

ER -