Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the ow of the cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) the presence of cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) a nonlinear nonlocal free boundary condition that involves boundary curvature. We establish the bifurcation of traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove the existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation in a free boundary setting (which is obtained via a reduction of the original system).