# American Institute of Mathematical Sciences

December  2018, 11(6): 1359-1376. doi: 10.3934/krm.2018053

## On the convergence to critical scaling profiles in submonolayer deposition models

 1 Departamento de Ciȇncias e Tecnologia, Universidade Aberta, Lisboa, Portugal 2 Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal 3 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

* Corresponding author: F.P. da Costa

Received  July 2017 Published  June 2018

Fund Project: Partially funded by FCT/Portugal through project RD0447/CAMGSD/2015.

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 $n≥ 2$ 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 $x = τ$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $ξ : = (x-\tau )/\sqrt \tau$, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $Φ_{2,n}(ξ)$ when $x, \tau \to +∞$ with $ξ$ fixed, as well as the rate at which the limit is approached.

Citation: Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
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##### References:
Lines with $\xi = \text{constant}$ (full), and with $\eta = \text{constant}$ (dashed) in the $(j,\tau)$ plane. The values used for these parameters are the following, in counterclockwise direction: $\xi = 5.0, 2.0, 1.0, 0.5, 0.0, -0.3, -0.5, -0.7, -0.9,$ and $\eta = 1/4,$ $1/3,$ $1/2,$ $1,$ $2,$ $3,$ $4$.
Graph of the similarity profile $\Phi_{2,n}(\xi)$ for different values of $n$.
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