Sharp interface limit in a phase field model of cell motility

Leonid Berlyand; Mykhailo Potomkin; Volodymyr Rybalko; Leonid Berlyand; Mykhailo Potomkin; Volodymyr Rybalko

doi:10.3934/nhm.2017023

Networks and Heterogeneous Media

2017, Volume 12, Issue 4: 551-590. doi: 10.3934/nhm.2017023

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Sharp interface limit in a phase field model of cell motility

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2.
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine, 47 Nauky Ave., 61103 Kharkiv, Ukraine

Received: 01 March 2017 Revised: 01 August 2017
Fund Project: The work of LB was supported by NSF grants DMS-1106666 and DMS-1405769. The work of VR and MP was partially supported by NSF grant DMS-1106666.
35Q92, 35K51, 35B25

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic behavior 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 nonlinear 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 parameter. 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.
- Allen-Cahn equation,
- phase field model,
- sharp interface limit,
- cell motility,
- traveling wave
Citation: Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko. Sharp interface limit in a phase field model of cell motility[J]. Networks and Heterogeneous Media, 2017, 12(4): 551-590. doi: 10.3934/nhm.2017023

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Abstract

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic behavior 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 nonlinear 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 parameter. 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.

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