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

Abstract

This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_tφ+ (-Δ+1)(a(\cdot)φ-J\astφ+ G'(φ)) = 0 \quad \mbox{in}\ Ω\times(0, T) $$ in an unbounded 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 the case that $\Omega$ is a bounded domain and $-\Delta +1$ is replaced with $-\Delta$, this problem has been studied by using a Faedo--Galerkin approximation 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 and energy estimates of weak solutions for the above problem on an unbounded domain.