  This paper develops an abstract theory for subdifferential operators to give
existence and uniqueness of solutions to the initial-boundary problem (P) for
the nonlinear diffusion equation in an unbounded domain
$\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as
  \[
  \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u)
  = g \quad \mbox{in}\ \Omega\times(0, T),
  \] which represents the porous media, the fast diffusion equations, etc.,
where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and
$T>0$. Existence and uniqueness for (P) were directly proved under a growth
condition for $\beta$ even though the Stefan problem was excluded from examples
of (P). This paper completely removes the growth condition for $\beta$ by
confirming Cauchy's criterion for solutions of the following approximate
problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$:
  \[
  \frac{\partial u_{\varepsilon}}{\partial t}
  + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon})
+ \pi_{\varepsilon}(u_{\varepsilon}))
  = g \quad \mbox{in}\ \Omega\times(0, T),
  \] which is called the Cahn--Hilliard system, even if $\Omega \subset
\mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be
seen that the Stefan problem is covered in the framework of this paper.
