  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.

Classical Analysis and ODEs