Asymptotic behavior of the generalized Becker-D\"oring equations for general initial data

TL;DR

This paper analyzes the long-term behavior of solutions to the generalized Becker-Döring equations, showing convergence to equilibrium states depending on initial density and establishing new estimates under broad initial conditions.

Contribution

It extends previous results by proving asymptotic behavior for general initial data under a detailed balance assumption, including cases with densities above the critical density.

Findings

Solutions with initial density ≤ critical density converge strongly to equilibrium.

Solutions with initial density > critical density converge weakly to the critical equilibrium.

A new tail estimate for solutions below the critical density was developed.

Abstract

We prove the following asymptotic behavior for solutions to the generalized Becker-D\"oring system for general initial data: under a detailed balance assumption and in situations where density is conserved in time, there is a critical density $\rho_s$ such that solutions with an initial density $\rho_0 \leq \rho_s$ converge strongly to the equilibrium with density $\rho_0$, and solutions with initial density $\rho_0 > \rho_s$ converge (in a weak sense) to the equilibrium with density $\rho_s$. This extends the previous knowledge that this behavior happens under more restrictive conditions on the initial data. The main tool is a new estimate on the tail of solutions with density below the critical density.