*Accepted Paper*

**Inserted:** 19 apr 2017

**Last Updated:** 18 aug 2018

**Journal:** Commun. Math. Sci.

**Year:** 2017

**Abstract:**

We consider the evolution equation
\begin{equation}\label{abs1}
h_{t=\ddt \F^{-1}(-aE \F(h)) - r/h^2 -\ddt h ,
}
\end{equation}
introduced in {\cite{TS}} by Tekalign and Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic
substrate. In the expression above, $h$ denotes the surface height of the film,
$\F$ is the Fourier transform,
and $a$, $E$, $r$ are positive material constants. For simplicity, we set
$aE=r=1$.
As this equation does not have any particular structure, its analysis is quite challenging. Therefore, we introduce
the auxiliary equation (with $c$ being a given constant)
\begin{equation}\label{abs2}
u_{t=\gr - \div u - (\div u+c)^{-2} -\ddt \div u ,
}
\end{equation}
which has a variational structure. Equivalency between \eqref{abs1} and \eqref{abs2} will hold under sufficient
regularity on the solution.
The main aim of this paper is to provide an analytical validation
to \eqref{abs2}, by proving existence and regularity properties for
weak solutions, under suitable assumptions on the initial datum.

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