Trend to equilibrium for discrete coagulation equations with strong fragmentation and without 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 an 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 exponential.