Global stability for solutions to the exponential PDE describing epitaxial growth

Abstract

In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=\Delta e^{-\Delta h}$ in the whole space ${\mathbb{R}}_x^d$. We assume the initial data is of medium size in the Wiener algebra $A({\mathbb{R}}^d)$; we use the initial condition $\Delta h_0\in A({\mathbb{R}}^d)$ which is scale-invariant with respect to the invariant scaling of the exponential PDE. This exponential PDE was derived in [18] and more recently in [22].