Global stability for solutions to the exponential PDE describing epitaxial growth

TL;DR

This paper establishes the global well-posedness, decay rates, and analyticity of solutions to a nonlinear exponential PDE modeling epitaxial growth, under medium-sized initial conditions in a critical function space.

Contribution

It provides the first comprehensive proof of global existence, decay, and analyticity for this PDE in the whole space, extending previous partial results.

Findings

Global existence and uniqueness of solutions

Optimal large-time decay rates established

Uniform analyticity gain proven

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 $\mathbb{R}^d_x$. 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).