### Abstract

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.

Original language | English |
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Pages (from-to) | 657-692 |

Number of pages | 36 |

Journal | Communications in Mathematical Physics |

Volume | 104 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1986 |