Dynamics of the modified viscous Cahn-Hilliard equation in \(\mathbb{R}^N\)
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.
scientific article; zbMATH DE number 5794087