Analysis of a Nonlocal Model for Spontaneous Cell Polarization
Abstract
In this work, we investigate the dynamics of a nonlocal model describing spontaneous cell polarization. It consists of a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviors in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller–Segel system, but critical spaces are different ($L^N$ instead of $L^{N/2}$ due to the coupling on the boundary). In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges toward a nontrivial stationary configuration.
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Copyright © 2012 Society for Industrial and Applied Mathematics.
History
Submitted: 23 May 2011
Accepted: 25 January 2012
Published online: 12 April 2012
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