Strong solutions to a fourth order exponential PDE describing epitaxial growth

TL;DR

This paper proves the global existence of strong solutions for a fourth order exponential PDE modeling epitaxial crystal growth, introducing a novel analytical approach to handle exponential nonlinearities.

Contribution

The paper presents a new method for establishing global solutions to an exponential PDE, expanding understanding of nonlinear PDEs in epitaxial growth models.

Findings

Established global existence of strong solutions.

Identified key initial data controls for solutions.

Introduced a novel analytical approach for exponential PDEs.

Abstract

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation $\partial_tu-\Delta e^{-\Delta u}+e^{-\Delta u}-1=0$. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects \cite{GLLM}. Our investigations reveal that we must control the size of both $\left\| e^{-\Delta u(x,0)}\right\|_{W^{2,2}(\Omega)}$ and $\left\| e^{\Delta u(x,0)}\right\|_{\infty,\Omega}$ suitably to achieve our results. Related results in \cite{GM,LS} were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity.