The Neumann boundary problem for a nonlocal Cahn-Hilliard equation
type: Item
instance of: scholarly article
title: The Neumann boundary problem for a nonlocal Cahn-Hilliard equation
zbMATH Open document ID: 1081.45004
author: Peter W. Bates
author: Jianlong Han
published in: Journal of Differential Equations
publication date: 2005-06-01T00:00:00Z
review text: The authors consider the following nonlocal Cahn-Hilliard equation: \[ \frac{\partial u}{\partial t} = \Delta\left(\int_\Omega J(x-y)dyu(x) -\int_\Omega J(x-y)u(y)dy + f(u)\right), \quad \text{in } \Omega, \;t>0, \] \[ \frac {\partial\left(\int_\Omega J(x-y)dyu(x)-\int_\Omega J(x-y)u(y)dy + f(u)\right) }{\partial n} = 0, \quad\text{on } \partial\Omega, \;t>0,\tag{1} \] \[ u(x,0) = u_0 (x), \] where \(0\leq J(x)=J(-x), T>0, \) \(\Omega \subset \mathbb{R}^n\) is bounded, and \(f\) is bistable. The authors prove the existence and uniqueness of solution to problem (1) as well as the continuous dependence on the initial data. Furthermore, they prove the existence of absorbing sets and apply their results to other nonlocal problems.
reviewed by: Mohamed O. El-Doma
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 2173045
zbMATH Keywords: integro-partial differential equation
zbMATH Keywords: continuous dependence on initial data
zbMATH Keywords: absorbing set
zbMATH Keywords: a priori estimate
zbMATH Keywords: nonlocal Cahn-Hilliard equation
zbMATH Keywords: phase transition
zbMATH Keywords: long-range interaction
zbMATH Keywords: nonlinear Poincaré inequality
MaRDI profile type: Publication
full work available at URL: j.jde.2004.07.003
OpenAlex ID: W2003515679
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DOI: 10.1016/J.JDE.2004.07.003
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description: scientific article; zbMATH DE number 2173045