A Becker–Döring model with injection and irreversible fragmentation

Abstract We introduce and analyse a variant of the Becker–Döring equations that models the growth of clusters through the gain or loss of monomers. Motivated by enzymatic reactions in biology, this model incorporates irreversible fragmentation and monomers injection. We establish the well-posedness of the equations under suitable conditions on the kinetic rates. Then, as in the Becker–Döring equations, we distinguish two cases for the long-time behaviour of our solution; however, the distinction is made from the constant rate injection of monomers. While under strong fragmentation rate the system may exhibit infinite steady states, we prove that for low injection rate and moderate fragmentation, the solution converges locally exponentially fast to the steady state. Finally, we present an efficient scheme that preserves the asymptotic and allows fast computation by sub-sampling the clusters.