  We consider a system of two PDEs arising in modeling of motility of
eukariotic cells on substrates. This system consists of the Allen-Cahn equation
for the scalar phase field function coupled with another vectorial parabolic
equation for the orientation of the actin filament network. The two key
properties of this system are (i) presence of gradients in the coupling terms
(gradient coupling) and (ii) mass (volume) preservation constraints. We first
prove that the sharp interface property of initial conditions is preserved in
time. Next we formally derive the equation of the motion of the interface,
which is the mean curvature motion perturbed by a nonlinear term that appears
due to the properties (i)-(ii). This novel term leads to surprising features of
the the motion of the interface. Because of these properties maximum principle
and classical comparison techniques do not apply to this system. Furthermore,
the system can not be written in a form of gradient flow, which is why recently
developed Gamma-convergence techniques also can not be used for the
justification of the formal derivation. Such justification is presented in a
one-dimensional model problem and it leads to a stability result in a class of
'sharp interface' initial data.

Analysis of PDEs