# American Institute of Mathematical Sciences

December  2017, 12(4): 551-590. doi: 10.3934/nhm.2017023

## 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

* Corresponding author: Mykhailo Potomkin, mup20@ucs.psu.edu

Received  March 2017 Revised  August 2017 Published  October 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

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.

Citation: Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko. Sharp interface limit in a phase field model of cell motility. Networks & Heterogeneous Media, 2017, 12 (4) : 551-590. doi: 10.3934/nhm.2017023
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Illustration of the ansatz (34). Function $\rho_\varepsilon$ decays to a non-zero constant of the order $\varepsilon$ for $x\to\pm\infty$ and to a constant slightly different from 1 for $-a\leq x\leq a$ (solid). Dashed line represents the limiting profile, which is the characteristic function of $(-a, a)$
Left: $\Phi_\beta(V)$ for $\beta=150$ and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$, positive slope illustrates $\Phi'_\beta(0)>0$; Right: $\theta_0$, standing wave for the Allen-Cahn equation for $W_{\text{sym}}(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$ (dashed) and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$ (solid)
Left: Plot of function $\Phi_\beta(V)$ for $\beta =150>\beta_{\text{cr}}$; {\it Right}: Plot $c_0V- \Phi_\beta(V)$ for $\beta=150$ vs $F$. For $-F=1.5$ there is one intersection ((61) has one root). For each $-F=1.762$ and $-F=2.264$ there are two intersections ((61) has two roots). For $-F=2$ there are three intersections ((61) has three roots)
Sketch of the function $F(V)=-c_0V+\Phi_\beta (V)$; $F(V)$ has one local minimum, $F_{\text{min}}=F(V_{\text{min}})$, and one local maximum, $F_{\text{max}}=F(V_{\text{max}})$. Left: Until $F<F_{\max}$ we stay on the left branch. When $F$ exceeds $F_{\max}$ we jump on the right branch; Right: Until $F>F_{\min}$ we stay on the right branch; When $F$ becomes less than $F_{\min}$ we jump on the left branch. Red arrows on both figures illustrate jumps in velocities.
Hysteresis loop in the problem of cell motility. Simulations of $V=V(F)$ Left: (61) Jumping from the left to the right branches and back; Right: PDE system (57)-(58). On both figures arrows show in what direction the system $(V(t), F(t))$ evolves as time $t$ grows; red curve is for $F_{\uparrow}(t)$, blue curve is for $F_{\downarrow}(t)$
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