  We study a non-linear and non-local evolution equation for curves obtained as
the sharp interface limit of a phase-field model for crawling motion of
eukaryotic cells on a substrate. We establish uniqueness of solutions to the
sharp interface limit equation in the so-called subcritical parameter regime.
The proof relies on a Gr\"{o}nwall estimate for a specially chosen weighted
$L^2$ norm. Next, as persistent motion of crawling cells is of central interest
to biologists we study the existence of traveling wave solutions. We prove that
traveling wave solutions exist in the supercritical parameter regime provided
the non-linear term of the sharp interface limit equation possesses certain
asymmetry (related, e.g., to myosin contractility). Finally, we numerically
investigate traveling wave solutions and simulate their dynamics. Due to
non-uniqueness of solutions of the sharp interface limit equation we simulate a
related, singularly perturbed PDE system which is uniquely solvable. Our
simulations predict instability of traveling wave solutions and capture both
bipedal wandering cell motion as well as rotating cell motion; these behaviors
qualitatively agree with recent experimental and theoretical fidings.
