Kinetic behaviour of irreversible aggregation–annihilation process with input term

We study the kinetic behaviours of irreversible aggregation–annihilation processes with input of multi-polymers. In the models, an irreversible aggregation reaction occurs between any two clusters of the same species and an irreversible annihilation reaction occurs between two different species; meanwhile, the clusters of any size are gradually injected into the system. In a mean-field limit, we obtain the general solutions of the cluster-mass distributions for the cases with arbitrary input probability ρ(t). The results indicate that the cluster-mass distribution of either species satisfies a scaling law. Moreover, we also analyse emphatically the evolution behaviour of the total mass of either species under the condition of input probability ρ(t)=u(t+t0)v