Abstract
Initially a reaction-diffusion system with non-local reaction terms is build up as a mathematical model to illustrate the evolution of protein dimers within human cells. The derived system inspects the situation when chemical reactions occur when the two chemicals within a cell are in distance R, where such a distance is the reaction radius. Next, the long-time behavior of the solutions of the preceding non-local system is investigated as well as the phase separation phenomenon occurring when the reaction takes place very fast is also examined. It is actually shown that a two-phase Stefan problem is derived in the limit of infinite chemical reaction rate. Next the convergence of the global-in-time solution to the preceding system towards the unique stationary solution is derived. The chapter closes with some results on the determination of the decay rate of the above convergence towards the unique stationary solution.
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