Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

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

Pages: 735 – 762



Leonid Berlyand (Pennsylvania State University, University Park, Pa., U.S.A.)

Jan Fuhrmann (Institute of Mathematics, Johannes Gutenberg University, Mainz, Germany)

Volodymyr Rybalko (B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine)


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 flow 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).


traveling waves, free boundary, cell motility

2010 Mathematics Subject Classification

35B32, 35C07, 35R35, 92C17

Received 29 September 2017

Received revised 27 January 2018

Accepted 27 January 2018

Published 30 August 2018