article · 12/04/2012
ANALYSIS OF A NONLOCAL MODEL FOR SPONTANEOUS CELL POLARIZATION
Résumé
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.
Citer cet article
Calvez, V., Hawkins, R.-J., Meunier, N., & Voituriez, R. (2012). ANALYSIS OF A NONLOCAL MODEL FOR SPONTANEOUS CELL POLARIZATION. SIAM JOURNAL ON APPLIED MATHEMATICS, 72(2). https://doi.org/10.1137/11083486X@article{Calvez2012_253,
author = {Calvez, Vincent and Hawkins, Rhoda J. and Meunier, Nicolas and Voituriez, Raphael},
year = {2012},
month = {4},
title = {ANALYSIS OF A NONLOCAL MODEL FOR SPONTANEOUS CELL POLARIZATION},
journal = {SIAM JOURNAL ON APPLIED MATHEMATICS},
volume = {72},
number = {2},
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.},
url = {http://www.dx.doi.org/10.1137/11083486X},
doi = {10.1137/11083486X},
issn = {0036-1399},
}TY - JOUR
AU - Calvez, Vincent
AU - Hawkins, Rhoda J.
AU - Meunier, Nicolas
AU - Voituriez, Raphael
PY - 2012
DA - 2012/04/12
TI - ANALYSIS OF A NONLOCAL MODEL FOR SPONTANEOUS CELL POLARIZATION
JO - SIAM JOURNAL ON APPLIED MATHEMATICS
VL - 72
IS - 2
SN - 0036-1399
AB - 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.
DO - 10.1137/11083486X
UR - http://www.dx.doi.org/10.1137/11083486X
ER -