On the convergence to critical scaling profiles in submonolayer deposition models

TL;DR

This paper investigates the rate at which solutions of a mean field model for submonolayer atom deposition converge to critical scaling profiles, focusing on an inner expansion near a critical direction and establishing convergence and rate results.

Contribution

It introduces a new similarity variable for analyzing convergence to critical profiles and proves convergence and rate results in a mean field deposition model.

Findings

Proves convergence of solutions to a specific similarity profile

Establishes the rate at which the convergence occurs

Analyzes the inner expansion near the critical direction

Abstract

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $n\geq 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $x=\tau$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $\xi := (x-\tau)/\sqrt{\tau}$, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $\Phi_{2,n}(\xi)$ when $x, \tau\to +\infty$ with $\xi$ fixed, as well as the rate at which the limit is approached.