Strong solutions to a fourth order exponential PDE describing epitaxial growth

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 ${\partial }_{t}u-\Delta e^{-\Delta u}+e^{-\Delta u}-1=0$. 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 ${\Vert e^{-\Delta u(x,0)}\Vert }_{W^{2,2}(\Omega )}$ and ${\Vert e^{\Delta u(x,0)}\Vert }_{\infty ,\Omega }$ 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.