A PARABOLIC FREE BOUNDARY PROBLEM ARISING IN A MODEL OF CELL POLARIZATION

The amplification of an external signal is a key step in direction sensing of biological cells. We consider a simple model for the response to a time-depending signal, which was previously proposed by the last three authors. The model consists of a bulk-surface reaction-diffusion model. We prove that in a suitable asymptotic limit the system converges to a bulk-surface parabolic obstacle-type problem. For this model and a reduction to a nonlocal surface equation we show an L-1-contraction property and, in the case of time-constant signals, the stability of stationary states.