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This paper considers the initial-boundary value problem for the nonlocalCahn--Hilliard equation $$ \partial_t\varphi + (-\Delta+1)(a(\cdot)\varphi-J\ast\varphi + G'(\varphi)) = 0 \quad \mbox{in}\ \Omega\times(0, T) $$ in anunbounded domain $\Omega \subset \mathbb{R}^N$ with smooth bounded boundary,where $N\in\mathbb{N}$, $T>0$, and $a(\cdot), J, G$ are given functions. In thecase that $\Omega$ is a bounded domain and $-\Delta+1$ is replaced with$-\Delta$, this problem has been studied by using a Faedo--Galerkinapproximation scheme considering the compactness of the Neumann operator$-\Delta+1$ (cf. Colli--Frigeri--Grasselli (2012), Gal--Grasselli (2014)).However, the compactness of the Neumann operator $-\Delta+1$ breaks down when$\Omega$ is an unbounded domain. The present work establishes existence andenergy estimates of weak solutions for the above problem on an unboundeddomain.

Existence and energy estimates of weak solutions for nonlocal Cahn--Hilliard equations on unbounded domains