Strong solutions to a fourth order exponential PDE describing epitaxial growth

2021/06/28 by Brock C. Price, Xiangsheng Xu, Price, Brock C. +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Navier-Stokes equation solutions #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2106.14939

openalex publication_date 2021/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∂_tu-Δe^-Δu+e^-Δu-1=0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects \citeGLLM. Our investigations reveal that we must control the size of both ‖ e^-Δu(x,0)‖_W^2,2(Ω) and ‖ e^Δu(x,0)‖_∞,Ω suitably to achieve our results. Related results in \citeGM,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.

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