Asymptotic analysis for Cahn--Hilliard type phase field systems related to tumor growth in general domains

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.