  We consider a system of two coupled parabolic PDEs introduced in [1] to model
motility of eukaryotic cells. 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 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 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. Also, in the 1D
case we establish nontrivial traveling waves when the coupling parameter is
large enough.
