Sharp interface limit in a phase field model of cell motility

We consider a system of two PDEs introduced in [1] to model motility of eukaryotic cells on substrates. This system consists of the volume preserving Allen-Cahn equation for the scalar phase field function coupled with another parabolic equation for the orientation vector. We study the asymptotic behavior of solutions in the limit of a small parameter related to the width of the interface in phase field function (sharp interface limit). We prove that solutions do not blow up on finite time intervals and the sharp interface property of initial conditions is preserved in time. Next we formally derive the equation of motion of the interface, which is mean curvature motion with an additional nonlinear term. In the 1D case we establish nontrivial traveling waves which appear when the potential in the equation for phase field function has certain asymmetry and the parameter characterizing the strength of coupling is large enough. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special form of asymptotic expansion is introduced to reduce analysis to a single nonlinear PDE. 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.