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The amplification of an external signal is a key step in direction sensing ofbiological cells. We consider a simple model for the response to atime-depending signal, which was previously proposed by the last three authors.The model consists of a bulk-surface reaction-diffusion model. We prove that ina suitable asymptotic limit the system converges to a bulk-surface parabolicobstacle type problem. For this model and a reduction to a nonlocal surfaceequation we show an L1-contraction property and, in the case of time-constantsignals, the stability of stationary states.

A parabolic free boundary problem arising in a model of cell polarization