# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021194
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## Self polarization and traveling wave in a model for cell crawling migration

 1 Map5, UMR 8145 CNRS, Université de Paris, France 2 Department of Mathematics, University of Maryland, College Park MD 20742 USA 3 LaMME, UMR 8071 CNRS, Université Évry Val d'Essonne, France

Received  April 2021 Revised  September 2021 Early access December 2021

Fund Project: The second author is supported by NSF grant DMS-2009236

In this paper, we prove the existence of traveling wave solutions for an incompressible Darcy's free boundary problem recently introduced in [6] to describe cell motility. This free boundary problem involves a nonlinear destabilizing term in the boundary condition which describes the active character of the cell cytoskeleton. By using two different methods, a constructive method via a graph analysis and a local bifurcation method, we prove that traveling wave solutions exist when the destabilizing term is strong enough.

Citation: Alessandro Cucchi, Antoine Mellet, Nicolas Meunier. Self polarization and traveling wave in a model for cell crawling migration. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021194
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A shape of traveling wave solution $\Omega_0$ defined by (3.1) for $\beta/\lambda = 4$ and $\gamma = 1$. The red dots indicate the point $x_L<0$ on the left and $x_R>0$ on the right. The function $h(x)$ is defined for $x \in [x_L,x_R]$ such that conditions (3.2) – (3.3) hold. The graph of $h$ lies on the $y$ -positve part of the plane, while the graph of $-h$ on the $y$ -negative part