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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
DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n3.a6
Authors
Abstract
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).
Keywords
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