Existence Theorems for a Multidimensional Crystal Surface Model

Jian-Guo Liu Duke University Xiangsheng Xu Mississippi State University

Analysis of PDEs mathscidoc:1702.03019

SIAM Journal on Mathematical Analysis, 48, (6), 3667-3687, 2016.6
In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.
existence, nonlinear fourth-order parabolic equations, nonlinear functions of distributions, crystal surface model
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  title={Existence Theorems for a Multidimensional Crystal Surface Model},
  author={Jian-Guo Liu, and Xiangsheng Xu},
  booktitle={SIAM Journal on Mathematical Analysis},
Jian-Guo Liu, and Xiangsheng Xu. Existence Theorems for a Multidimensional Crystal Surface Model. 2016. Vol. 48. In SIAM Journal on Mathematical Analysis. pp.3667-3687. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207203802505518260.
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