Dynamics of a polymerization model on a graph

Abstract

This work is concerned with the dynamics of a polymerization process coupled with mass transfer and monomers injection, modeled by means of an infinite-dimensional system of Smoluchowski's equations in a finite graph. Under suitable assumptions on the system's aggregation coefficients, we show that, as a consequence of the injection mechanism, a sizable depletion of the pool of available reacting substances  occurs at some finite time, that can be estimated in terms of the parameters of the problem. By analogy with well-known results in chemical engineering, we interpret that result as the onset of a sol-gel phase transition. We suggest that this property might have some interest in the mathematical modeling of neurodegenerative processes, where the polymerization of some soluble proteins and their eventual aggregation into insoluble plaques play a remarkable role, which is not well understood as yet.