Analysis of the viscous Cahn–Hilliard equation in RN

Solvability of Cauchyʼs problem in RN for an extended viscous Cahn–Hilliard equation is studied. The problem is considered first in a standard Sobolev space H1(RN), next a notion of the ‘H-solution’ is introduced that is well adapted to the structure of the viscous Cahn–Hilliard equation. Several properties of the solutions are reported, in particular those connected with their asymptotic behavior. We collect here also known properties of the unbounded operator (−Δ)−1 in RN that are needed in our considerations.