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We study a two-dimensional free boundary problem that models motility ofeukaryotic cells on substrates. This problem consists of an elliptic equationdescribing the flow of cytoskeleton gel coupled with a convection-diffusion PDEfor the density of myosin motors. The two key properties of this problem are(i) presence of the cross diffusion as in the classical Keller-Segel problem inchemotaxis and (ii) nonlinear nonlocal free boundary condition that involvescurvature of the boundary. We establish the bifurcation of the traveling wavesfrom a family of radially symmetric steady states. The traveling waves describepersistent motion without external cues or stimuli which is a signature of cellmotility. We also prove existence of non-radial steady states. Existence ofboth traveling waves and non-radial steady states is established viaLeray-Schauder degree theory applied to a Liouville-type equation (which isobtained via a reduction of the original system) in a free boundary setting.

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