On the convergence to critical scaling profiles in submonolayer deposition models - Costa, Fernando Pestana da (2019)
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 $τ$ plane. In this paper we consider a different similarity variable, $ξ:= (x − τ )/ τ$ , corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $Φ_{2,n}(ξ)$ when $x, τ → +∞$ with $ξ$ fixed, as well as the rate at which the limit is approached.