Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition

The coagulation–fragmentation equation describes the concentration fi(t) of particles of size i ∈ N/{0} at time t ⩾ 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break–up. We show that when the rate of fragmentation is sufficiently stronger than that of coalescence, (fi(t))i ∈ N/{0} tends to a unique equilibrium as t tends to infinity. Although we suppose that the initial datum is sufficiently small, we do not assume a detailed balance (or reversibility) condition. The rate of convergence we obtain is, furthermore, exponential.