  We introduce a two-dimensional Keller-Segel type free boundary model for
motility of eukaryotic cells on substrates. The key ingredients of this model
are the Darcy law for overdamped motion of the cytoskeleton (active) gel and
Hele-Shaw type boundary conditions (Young-Laplace equation for pressure and
continuity of velocities). We first show that radially symmetric steady state
solutions become unstable and bifurcate to traveling wave solutions. Next we
establish linear and nonlinear stability of the steady states. We show that
linear stability analysis is inconclusive for both steady states and traveling
waves. Therefore we use invariance properties to prove nonlinear stability of
steady states.
