Dynamics of the modified viscous Cahn-Hilliard equation in \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: Dynamics of the modified viscous Cahn-Hilliard equation in \(\mathbb{R}^N\)
zbMATH Open document ID: 1223.35078
author: Chunyou Sun
author: Tomasz Dlotko
published in: Topological Methods in Nonlinear Analysis
publication date: 2010-10-01T00:00:00Z
review text: This paper studies the long term dynamics of a modified viscous Cahn-Hilliard equation of the form \[ (1-\nu)u_{t}=-\Delta (\Delta u + f(u)-\nu u_{t}), \quad \nu \in [0,1] \] in \(\mathbb R^N\). This equation, which finds applications in the study of binary alloys, interpolates between the Cahn-Hilliard equation \((\nu =0)\) and the semilinear heat equation \((\nu=1)\). For every \(\nu \in [0,1]\), the system is shown to be well-posed in the Sobolev space \(H^{1}(\mathbb R^N)\) and to possess a global attractor in this space which furthermore is upper semicontinuous at \(\nu=0\). The consideration of the equation in the whole of \(\mathbb R^N\) presents some technical difficulties due to the absense of compact embeddings of the relevant Sobolev spaces which are also attenuated by the lack of general maximum principles for fourth order equations. These are addressed by the use of proper a priori estimates including tail estimates which lead to the \((H^1,H^1)\) asymptotic compactness of the semigroup.
reviewed by: Athanasios Yannacopoulos
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K30
Mathematics Subject Classification ID: 37L15
zbMATH DE Number: 5794087
zbMATH Keywords: upper semicontinuous
zbMATH Keywords: lack of maximum principles
zbMATH Keywords: tail estimates
zbMATH Keywords: asymptotic compactness
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5794087