Global stability for solutions to the exponential PDE describing epitaxial growth

In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=\mathrm{\Delta }{e}^{-\mathrm{\Delta }h}$ in the whole space ${\mathbb{R}}_x^d$. We assume the initial data is of medium size in the critical Wiener algebra $\Delta h \in A($\mathbb{R}$^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).