Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type

Fourth-order semilinear parabolic equations of the Cahn-Hilliard-typeuare considered in a smooth bounded domain Ω ⊂ ℝuThe following three main problems are studied:(i) for the unstable model (0.1), with the −Δ(|u|(ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝ(iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.