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arXiv · 2106.14939

Strong solutions to a fourth order exponential PDE describing epitaxial growth

Brock C. Price·Xiangsheng Xu
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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.

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Brock C. Price, Xiangsheng Xu. 2021-06-28. Strong solutions to a fourth order exponential PDE describing epitaxial growth. https://arxiv.org/abs/2106.14939

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