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In this article we establish the global well-posedness of a recent model proposed by Noguera, Fritz, Clément and Baronnet for simultaneously describing the process of nucleation, growth and ageing of particles in thermodynamically closed, initially supersaturated systems. This model, which applies to precipitation in solution, vapor condensation and crystallization from a simple melt, can be seen as a highly nonlinear age-dependent population problem involving a delayed birth process and a hysteresis damage operator.