The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density ?{variant} is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions by c^(?{variant}), 0 ? ?{variant} ? ?{variant}_s, the principal result is that if the initial density ?{variant}₀ ? ?{variant}_s then the solution converges strongly to c^{(?{variant}o)}, while if ?{variant}₀ > ?{variant}_s the solution converges weak* to c^{(?{variant}s)}. In the latter case the excess density ?{variant}₀-?{variant}_s corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions. © 1986 Springer-Verlag.