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

Abstract

Abstract Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type ut + Δ2u = γu ± Δ(|u|p−1u) in Ω × ℝ+, (0.1) are considered in a smooth bounded domain Ω ⊂ ℝN with Navier-type boundary conditions on ∂Ω, or Ω = ℝN, where p > 1 and γ are given real parameters. The sign “+” in the “diffusion term” on the right-hand side means the stable case, while “−” reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for γ = 0, ut + Δ2u = ±Δ(|u|p−1u) in ℝN × R+. (0.2) The following three main problems are studied: (i) for the unstable model (0.1), with the −Δ(|u|p−1u), existence and multiplicity of classic steady states in Ω ⊂ ℝN and their global behaviour for large γ > 0; (ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝN × R+ for bounded initial data u0(x) in the subcritical case ; and (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.