Sharp interface limit in a phase field model of cell motility

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.