Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition

TL;DR

This paper analyzes the convergence rates to scaling profiles in a coagulation model for submonolayer deposition, revealing how initial conditions influence the approach rate despite loss of initial memory in the limit.

Contribution

It introduces a novel method to determine convergence rates to similarity profiles and shows how initial large cluster tail information affects the approach rate.

Findings

Established explicit convergence rates to similarity profiles.

Demonstrated preservation of initial large cluster tail information in approach rates.

Used center manifold and asymptotic methods for detailed analysis.

Abstract

We establish rates of convergence of solutions to scaling (or similarity) profiles in a coagulation type system modelling submonolayer deposition. We prove that, although all memory of the initial condition is lost in the similarity limit, information about the large cluster tail of the initial condition is preserved in the rate of approach to the similarity profile. The proof relies in a change of variables that allows for the decoupling of the original infinite system of ordinary differential equations into a closed two-dimensional nonlinear system for the monomer--bulk dynamics and a lower triangular infinite dimensional linear one for the cluster dynamics. The detailed knowledge of the long time monomer concentration, which was obtained earlier by Costin et al. in (O. Costin, M. Grinfeld, K.P. O'Neill and H. Park, Long-time behaviour of point islands under fixed rate deposition, Commun. Inf. Syst. 13, (2), (2013), pp.183-200) using asymptotic methods and is rederived here by center manifold arguments, is then used for the asymptotic evaluation of an integral representation formula for the concentration of $j$-clusters. The use of higher order expressions, both for the Stirling expansion and for the monomer evolution at large times allow us to obtain, not only the similarity limit, but also the rate at which it is approached.