### 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 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).

Original language | English (US) |
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Pages (from-to) | 735-762 |

Number of pages | 28 |

Journal | Communications in Mathematical Sciences |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - 2018 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications in Mathematical Sciences*,

*16*(3), 735-762. https://doi.org/10.4310/CMS.2018.v16.n3.a5