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In this paper we are concerned with the existence of a weak solution to theinitial boundary value problem for the equation $\frac{\partial u}{\partial t}= \Delta\left(\Delta u\right)^{-3}$. This problem arises in the mathematicalmodeling of the evolution of a crystal surface. Existence of a weak solution$u$ with $\Delta u\geq 0$ is obtained via a suitable substitution. Ourinvestigations reveal the close connection between this problem and theequation $\partial_t\rho+\rho^2\Delta^2\rho^3=0$, another crystal surface modelfirst proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in Physica D:Nonlinear Phenomena, bf 240 (2011), no. 21, 1771-1784.

Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions