  This article considers a limit system by passing to the limit in the
following Cahn--Hilliard type phase field system related to tumor growth as
$\beta\searrow0$:
  \begin{equation*}
  \begin{cases}
  \alpha\partial_{t} \mu_{\beta} + \partial_{t}
\varphi_{\beta}-\Delta\mu_{\beta} = p(\sigma_{\beta} - \mu_{\beta}) &
\mbox{in}\ \Omega\times(0, T),
  \\[1mm]
  \mu_{\beta} = \beta\partial_{t} \varphi_{\beta} + (-\Delta+1)\varphi_{\beta}
+ \xi_{\beta} + \pi(\varphi_{\beta}),\ \xi_{\beta} \in B(\varphi_{\beta})
  & \mbox{in}\ \Omega\times(0, T),
  \\[1mm]
  \partial_{t} \sigma_{\beta} -\Delta\sigma_{\beta} = -p(\sigma_{\beta} -
\mu_{\beta}) & \mbox{in}\ \Omega\times(0, T)
  \end{cases} \end{equation*} in a bounded or an unbounded domain $\Omega
\subset \mathbb{R}^{N}$ with smooth bounded boundary. Here $N\in\mathbb{N}$,
$T>0$, $\alpha>0$, $\beta>0$, $p\geq0$, $B$ is a maximal monotone graph and
$\pi$ is a Lipschitz continuous function. In the case that $\Omega$ is a
bounded domain, $p$ and $-\Delta+1$ are replaced with $p(\varphi_{\beta})$ and
$-\Delta$, respectively, and $p$ is a Lipschitz continuous function,
Colli--Gilardi--Rocca--Sprekels (2017) have proved existence of solutions to
the limit problem with this approach by applying the Aubin--Lions lemma for the
compact embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ and the continuous
embedding $L^2(\Omega) \hookrightarrow (H^1(\Omega))^{*}$. However, the
Aubin--Lions lemma cannot be applied directly when $\Omega$ is an unbounded
domain. The present work establishes existence of weak solutions to the limit
problem both in the case of bounded domains and in the case of unbounded
domains. To this end we construct an applicable theory for both of these two
cases by noting that the embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is
not compact in the case that $\Omega$ is an unbounded domain.
